00000000

PhysicsD10K

Many Body PhysicsF7K0

An introduction to many body physics from the class PHY 905-302 at Michigan State University. This course follows the book: Introduction to Many Body Physics by Pures Coleman.

1 Quantum FieldsC98M

1.1 Introduction7M27

Definition 1.1.1  The many-body Schrödinger equation is a general quantum equation that can be used to describe the dynamics of the wave-function $\Psi(\vec{x}_1,\dots,\vec{x}_N)$ of any collection of $N$ particles $\vec{x}_1,\dots,\vec{x}_N$ with masses $m_j$, potentials $U_i(\vec{x}_i)$ and interaction potentials $V(\vec{x}_i-\vec{x}_j)$.
\[\left[\frac{-\hbar^2}{2}\sum_j\frac{\nabla_j^2}{m_j} + \sum_{i<j}V(\vec{x}_i-\vec{x}_j) + \sum_jU_i(\vec{x}_i)\right]\Psi = i\hbar\frac{\partial \Psi}{\partial t}\]

Definition 1.1.2  The Coulomb Potential $V(r)$ describes the potential felt between two charged particles with charge equal to the elementary charge $e$.
\[V(r) = \frac{e^2}{4\pi\varepsilon_0}\frac{1}{r}\]

While the many-body Shrodinger equation entirely describes all the dynamics of all metals, chemistry, and materials. It is impossible to solve for the number of particles in macroscopic systems, which is typically on the order of $10^{23}$.

1.2 Second Quantization28H5

Definition 1.2.1 

Definition 1.2.2 

Ultrafast Phenomena60N0

1 Introduction7NZC

1.1 SI Units8ZFC

Definition 1.1.1  The SI unit system is the most popular system of units that uses the fundamental units of seconds, meters, kilograms, ampere and Kelvin to derive a system of units to describe the universe.

Definition 1.1.2  A second (s) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

Definition 1.1.3  A meter (m) is the SI unit of distance that is exactly the distance light travels in $1/299792458$ seconds.

Definition 1.1.4  A kilogram (kg) is the SI unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−34}\text{kg } \text{m}^2\text{s}^{−1}$.

Definition 1.1.5  An ampere (A) is the SI unit of current that is exactly the flow of $10^{19}/1.602176634$ protons per second.

Definition 1.1.6  A Kelvin (K) is the SI unit of absolute temperature defined exactly by fixing Boltzmann's constant $k=1.380649\times 10^{-23}\text{kg }\text{m}^2 \text{s}^{-2}\text{K}^{-1}$.

Definition 1.1.7  A Coulomb (C) is the SI unit of charge defined by $\text{C} = \text{A s}$ or exactly $10^{19}/1.602176634$ protons.

Definition 1.1.8  A Newton (N) is the SI unit of force defined by $\text{N} = \text{kg m}/\text{s}^{2}$.

Definition 1.1.9  A Joule (J) is the SI unit of energy defined by $\text{J} = \text{N m} = \text{kg }\text{m}^{2}/\text{s}^{2}$.

Definition 1.1.10  A Watt (w) is the SI unit of power defined by $\text{w} = \text{J}/\text{s}$.

Definition 1.1.11  A Pascal (Pa) is the SI unit of pressure defined by $\text{Pa}=\text{N}/\text{m}^2 = \text{J}/\text{m}^3 = \text{kg }\text{m}^{-1}\text{s}^{-2}$.

Definition 1.1.12  A Volt (V) is the SI unit of electric potential defined by $\text{V} = \text{J}/\text{C} = \text{w}/\text{A} = \text{kg }\text{m}^2\text{s}^{-3}\text{A}^{-1}$.

Definition 1.1.13  A Volt per Meter (V/m) is the SI unit of electric field defined by $\text{V}/\text{m} = \text{N}/\text{C} = \text{kg m}\text{s}^{-2}\text{A}^{-1}$.

Definition 1.1.14  A Telsa (T) is the SI unit of magnetic field defined by $\text{T} = \text{V s}/\text{m}^2 = \text{kg }\text{s}^{-2}\text{A}^{-1}$.

Definition 1.1.15  The fine structure constant denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value¹:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

Law 1.1.16  Maxwell's Equations are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]\[\nabla\cdot \vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

Definition 1.1.17  The vacuum permittivity $\varepsilon_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\varepsilon_0 = \frac{e^2}{2\alpha hc}\]

Definition 1.1.18  The vacuum permeability $\mu_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\mu_0 = \frac{2\alpha h}{e^2c}\]

Result 1.1.19  The product of vacuum permittivity and vacuum permeability is the reciprocal of the speed of light squared.
\[\varepsilon_0\mu_0 = \frac{1}{c^2}\]

1.2 Timeline of an ExcitationD9FA

Definition 1.2.1  A band structure is a diagram describing the states of a system (usually a crystal) that plots the energy of the states at different momentum wave vectors $\vec{k}$. For most materials this splits the energy levels of the atoms in the lattice into continuous bands of states at different energy and momenta.

Definition 1.2.2  The conduction band in is the lowest energy band that is above the Fermi level.

Definition 1.2.3  The valence band is the highest energy band that is below the Fermi level.

Example 1.2.4  Consider the band structure of the semiconductor GaAs. It is common for the greek letter $\Gamma$ to mark $\vec{k}=0$. The conduction band has three valleys where electrons could oscillate. The left one is the L-valley the center one is the $\Gamma$-valley and the right one is the X-valley. For an oscillating electron in one of these energy valleys, it would experience an effective mass dependent on the curvature of the valley.

Definition 1.2.5  The effective mass $m^*$ is the apparent mass of an electron oscillating in a valley of the band structure of a material determined by the curvature of the band structure as described by the following equation.
\[m^* = \hbar^2\left( \frac{\partial^2 E}{\partial k^2} \right)^{-1}\]

Example 1.2.6  Some effective masses of electrons in GaAs are listed below. In GaAs these are significantly less than the mass of an electron in free space.

Definition 1.2.7  A quasiparticle is a group of particles that act collectively as if they were a single particle.

Definition 1.2.8  An electron quasiparticle is the quasiparticle that describes how an electron behaves due to the influence of the band structure.

Definition 1.2.9  An electron-hole quasiparticle is the quasiparticle that describes how a missing electron behaves due to the influence of the band structure.

Example 1.2.10  Consider the timeline of an excitation caused by photon absorption. Electron carry very little momentum so the excited electrons move almost strait upwards on the band diagram. The electrons and holes thermalize in 10-100fs to a typical temperature of 1000K. They experience cooling through electron phonon scattering in 100fs-1ps. Finally, they decay back towards equilibrium through electron-hole recombination in 100ps-1ns.

Ahmed, I., et al., Light Science & Applications 10(1), 174. (2021) 10.1038/s41377-021-00609-3

Definition 1.2.11  The photon absorption process is when particles (typically electrons) absorb photons and are excited to a higher energy. Since photons carry very little momentum the momentum of the particles does not change very much.

Definition 1.2.12  The thermalization process is when particles (typically electrons) scatter off each other broadening the energy spectrum into a Fermi-Dirac distribution.

Definition 1.2.13  The electron-phonon scattering process is when electrons scatter off lattice vibrations (phonons) until they reach thermal equilibrium with the lattice.

Definition 1.2.14  The electron-hole recombination process is when electrons and electron-hole quasiparticles recombine to annihilate each other. The rate of this process is determined by the overlap of there wave functions.

Definition 1.2.15  A direct band-gap semiconductor is a semiconductor where the highest energy region of the valence band and the lowest energy region of the conduction band are at the same momentum.

Definition 1.2.16  An indirect band-gap semiconductor is a semiconductor where the highest energy region of the valence band and the lowest energy region of the conduction band are at very different same momentum. The difference in momentum makes electron-hole recombination after absorption less lightly, which can vastly increase the lifetimes of an excitation.

1.3 Examples of Photoexcitation Effects0MWH

Definition 1.3.1  An in-gap state is a (usually spatially localized) state in the energy gap of a semiconductor.

Result 1.3.2  For semiconductors with in-gap states carriers can be trapped in these states at momentum values that affect the probability of electron-hole recombination.

Definition 1.3.3  The position diffusion coefficient denoted $D$ is the rate at which the standard deviation $\sigma \propto \sqrt{Dt}$ in position space of diffusing particles spreads out over time. When describing the scattering of particles with a Drude model, the diffusion coefficient is defined,
\[D = \frac{\tau k_BT}{m^*} = \tau v\]
where $\tau$ is the average scattering time, $T$ is temperature, $m^*$ is the effective mass and $v$ is the thermal velocity.

Definition 1.3.4  The photo-Dember effect is the emission of terahertz due to transient dipoles produced by diffusion of charge carriers after photo-excitation.

Definition 1.3.5  A solar cell is a device that uses photoexcitation to generate electricity. In a solar cell, the band structure causes the electrons and holes produced by photo excitation to move to opposite ends of the crystal thereby producing an voltage across the sample.

Definition 1.3.6  The Auger recombination process is when a third carrier allows for an electron and hole to recombine at different momenta, the extra momentum is transferred to the third particle.

Definition 1.3.7  The Frank Condon Effect is the production of coherent oscillations of photo excited carriers that is usually produced by a momentum difference between bands in strongly correlated systems or molecules.

1.4 Ultrafast LasersNHDH

Definition 1.4.1  The intensity denoted $I$ of light is the power transmitted per unit area. This is related to the electric field of a electromagnetic wave by the following equation.
\[I = \frac{1}{2}\varepsilon_0cE^2\]

Corollary 1.4.2  The electric field from intensity of an electromagnetic wave is given by $E = \sqrt{\frac{2I}{\varepsilon_0 c}}$.

2 Nonlinear OpticsTP9J

2.1 The Wave Equation from Maxwell's EquationsA9FW

Law 2.1.1  Gauss' law states that the divergence of the electric field is equal to the charge density divided by the permitivity of free space.
\[\vec\nabla\cdot\vec{E} = \rho/\varepsilon_0\]

Law 2.1.2  No magnetic monopoles law states that the divergence of the magnetic field is zero.
\[\vec{\nabla}\cdot\vec{B}=0\]

Law 2.1.3  Faraday's law states that the curl of the electric field $\vec{E}$ is equal to the negative time derivative of the magnetic field $\vec{B}$.
\[\vec\nabla\times\vec{E} = - \frac{\partial B}{\partial t}\]

Law 2.1.4  Ampere's law states that the curl of the magnetic field $\vec{B}$ is equal to the current density $\vec{J}$ plus the permittivity of free space $\varepsilon_0$ times the time derivative of the electric field $\vec{E}$ all multiplied by the permeability of free space $\mu_0$.
\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

These four laws together form Maxwell's equations.

Law 2.1.6  Maxwell's Equations are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]\[\nabla\cdot \vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

Result 2.1.7  The wave equation from currents and charge densities can be derived from Maxwell's equations by applying some vector identities. This describes electromagnetic waves produced by time time varying currents and charge densities.
\[\nabla^2\vec{E} - \mu_0\varepsilon_0\frac{\partial^2\vec{E}}{\partial t^2} = \mu_0\frac{\partial \vec{J}}{\partial t}+\vec{\nabla}\left(\frac{\rho}{\varepsilon_0}\right)\]

Corollary 2.1.8  The wave equation in free space immediately follows for constant current $\vec{J}$ and constant charge density $\rho$. This describes electromagnetic waves traveling in free space.
\[\nabla^2\vec{E} - \mu_0\varepsilon_0\frac{\partial^2\vec{E}}{\partial t^2} = 0\]

Result 2.1.9  The plane wave solution to the wave equation is given by the following electric field.
\[\vec{E}(\vec{r}) = E_0\hat{n}e^{i(\vec{k}\cdot\vec{r} - \omega t+\phi)}+E_0\hat{n}e^{ii(\vec{k}\cdot\vec{r} - \omega t+\phi)}\]

2.2 Fields and PolarizationH6WH

Definition 2.2.1  The electric field denoted $\vec{E}(\vec{r})$ is a vector field of the force that would be felt by a test charge at a point in space. The units of electric field are Newtons per Coulomb denoted $N/C$. For a charge $q$ at position $\vec{r}$ the force $\vec{F}$ from electric field $\vec{E}$ can be calculated with the following equation:
\[\vec{F} = q\vec{E}(\vec{r})\]

Definition 2.2.2  The electric displacement field denoted $\vec{D}$ is defined in terms of the electric field $\vec{E}$ the polarization $\vec{P}$.
\[\vec{D} = \varepsilon_0\vec{E} + \vec{P}\]

Result 2.2.3  The divergence of the displacement field $\vec{D}$ is the free charge density $\rho_f$.
\[\vec{\nabla}\cdot\vec{D} = \rho_f\]

Definition 2.2.4  The polarization denoted $\vec{P}$ is the electric dipole moment per unit volume of the bound charge density $\rho_b$ in a material.
\[\vec{P} = \frac{d\vec{p}}{dV}\]\[- \nabla\cdot\vec{P} = \rho_b\]

Definition 2.2.5  The electric susceptibility denoted $\chi_e$ of a material is the tensor that describes the polarization $\vec{P}$ of a material in response to an electric field $\vec{E}$, where $\varepsilon_0$ is the vacuum permittivity.
\[\vec{P} = \varepsilon_0\chi_e\vec{E}\]

Definition 2.2.6  The magnetic field or magnetic flux density denoted $\vec{B}$ is the vector field that describes the force per length of current in a region of space. This is the actual magnetic field at a point in space.

Definition 2.2.7  The magnetic field strength denoted $\vec{H}$ is the vector that described the external contribution to the magnetic field in a material not intrinsic to the material's magnetization $\vec{M}$, where $\mu_0$ is the vacuum permeability.
\[\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}\]

Result 2.2.8  The curl of the magnetic field strength $\vec{H}$ is the current density $\vec{J}$ plus the time derivative of the displacement field $\vec{D}$.
\[\nabla\times\vec{H} = \vec{J} + \frac{\partial D}{\partial t}\]

Definition 2.2.9  The magnetic susceptibility denoted $\chi_m$ is the tensor that describes the magnetization $\vec{M}$ produced my a material in response to magnetic field strength $\vec{H}$.
\[\vec{M} = \chi_m\vec{H}\]

Definition 2.2.10  The relative permittivity denoted $\varepsilon_r$ in a material is defined with the following relations, where $\chi_e$ is the electric susceptibility, $\varepsilon_0$ is the vacuum permittivity and $\varepsilon$ is the permittivity of the medium.
\[\varepsilon_r = 1+\chi_e,\quad \varepsilon_r = \varepsilon/\varepsilon_0,\quad \varepsilon=\varepsilon_r\varepsilon_0\]

Table 2.2.11  Relations between Dielectric Function, Conductivity and Index of Refraction - The following table describes the relationships between the complex dielectric function $\tilde{\varepsilon}$, conductivity $\tilde{\sigma}$ and index of refraction $\tilde{n}$.

2.3 Nonlinear Suseptability in 1D2N6T

Result 2.3.1  The wave equation from polarization can be derived from Maxwell's equations by applying some vector identities. This describes electromagnetic waves produced by time varying polarization $\vec{P}$.
\[\nabla^2\vec{E} - \frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}=\mu_0\frac{\partial^2\vec{P}}{\partial t^2}\]

Definition 2.3.2  The 1D nonlinear susceptibility is the Taylor expansion of the electric susceptibility in 1 dimension.
\[P = \varepsilon_0\left(\chi_e^{(1)}E+\chi_e^{(2)}E^2+\chi_e^{(3)}E^3+\chi_e^{(4)}E^4+\dots\right) = \varepsilon_0(\varepsilon_r^{(1)}-1)E + P^{(NL)}\]

Result 2.3.3  The 1D wave equation in a material can be derived by plugging the 1D nonlinear susceptibility into the wave equation from polarization. It describes the motion of light through a material with a source term generated from the nonlinear component of the polarization.
\[\frac{\partial^2E}{\partial t^2} - \mu_0\varepsilon\frac{\partial^2{E}}{\partial t^2}=\mu_0\frac{\partial^2P^{(NL)}}{\partial t^2}\]

Result 2.3.4  1D second order two pulse generation describes the second order polarization produced when waves of light at frequencies $\omega_1$ and $\omega_2$ propagate in a material with second order electric susceptibility $\chi_e^{(2)}$.
\[E(t) = E_1e^{-i\omega_1t} + E_1e^{i\omega_1t} + E_2e^{-i\omega_2t} + E_2e^{i\omega_2t}\]\[P^{(2)}(t) = \varepsilon_0\chi^{(2)}E^2(t)=\left(E_1e^{-i\omega_1t} + E_1e^{i\omega_1t} + E_2e^{-i\omega_2t} + E_2e^{i\omega_2t}\right)^2\]\[ = \varepsilon_0\chi^{(2)}\left(E_1^2 e^{-2i\omega_1t} + E_2^2 e^{-2i\omega_2t} + E_1^2 e^{2i\omega_1t} + E_2^2 e^{2i\omega_2t}\right.\]\[+2E_1E_2e^{-i(\omega_1+\omega_2)t} + 2E_1E_2e^{i(\omega_1+\omega_2)t} \]\[+2E_1E_2^*e^{-i(\omega_1-\omega_2)t} + 2E_1^*E_2e^{i(\omega_1-\omega_2)t} \]\[\left.+E_1E_1^* + E_2E_2^*\right)\]

Definition 2.3.5  The second harmonic generation process is the second order nonlinear process that produces light with twice the frequency of the incoming light. It is produced by the following terms of the polarization, when waves of light at frequencies $\omega_1$ and $\omega_2$ propagate in a material with second order electric susceptibility $\chi_e^{(2)}$.
\[\varepsilon_0\chi^{(2)}_e\left( E_1^2 e^{-2i\omega_1t} + E_2^2 e^{-2i\omega_2t} + E_1^2 e^{2i\omega_1t} + E_2^2 e^{2i\omega_2t} \right)\]

Definition 2.3.6  The sum frequency generation process is the second order nonlinear process that produces light with the sum frequency of the two incoming light waves. It is produced by the following terms of the polarization, when waves of light at frequencies $\omega_1$ and $\omega_2$ propagate in a material with second order electric susceptibility $\chi_e^{(2)}$.
\[\varepsilon_0\chi_e^{(2)}\left( 2E_1E_2e^{-i(\omega_1+\omega_2)t} + 2E_1E_2e^{i(\omega_1+\omega_2)t} \right)\]

Definition 2.3.7  The difference frequency generation process is the second order nonlinear process that produces light with the difference frequency of the two incoming light waves. It is produced by the following terms of the polarization, when waves of light at frequencies $\omega_1$ and $\omega_2$ propagate in a material with second order electric susceptibility $\chi_e^{(2)}$.
\[\varepsilon_0\chi_e^{(2)}\left( 2E_1E_2^*e^{-i(\omega_1-\omega_2)t} + 2E_1^*E_2e^{i(\omega_1-\omega_2)t} \right)\]

Definition 2.3.8  The optical rectification process is the second order nonlinear process that produces a DC polarization across the crystal. It is produced by the following terms of the polarization, in a material with second order electric susceptibility $\chi_e^{(2)}$.
\[2\varepsilon_0\chi_e^{(2)}\left( E_1E_1^* + E_2E_2^* \right)\]

Definition 2.3.9  The contracted d-tensor denoted $d_{i,j}$ is the contracted form of the second order term of the electric susceptability $\chi_e^{(2)}$.
\[d_{i,j} = \frac{1}{2}\chi_{i,j}^{(2)}\]

Definition 2.3.10  A 4-wave mixing process refers to the resulting $\chi_e^{(3)}$ processes produced by repeating the same mixing process described for 1D second order two pulse generation for a material with a non-zero $\chi_e^{(3)}$.

2.4 Lorentz and Drude Model718F

Definition 2.4.1  The Lorentz oscillator is a model for the polarization of bound electrons in a solid based on a charged mass $m$ with charge $q$, density $N$, a linear restoring force and a damping coefficient $\gamma$. The position of the charge in the potential well is $x(t)$.
\[\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + \omega_0^2x = \frac{q}{m}E(t)\]\[P(t) = Nqx(t)\]

Result 2.4.2  The plane wave solution to the Lorentz oscillator for position of the charge $x(t)$, polarization $P(t)$ is given by the following equations.
\[x(t) = \frac{qE_0}{m\left(-\omega^2 - i\gamma\omega +\omega_0^2\right)} e^{-i\omega t}\]\[P(t) = \frac{Nq^2E_0}{m\left(-\omega^2 - i\gamma\omega +\omega_0^2\right)} e^{-i\omega t}\]

Result 2.4.3  The plane wave permittivity of the Lorentz oscillator for permittivity $\varepsilon$, relative permittivity $\varepsilon_r$ and susceptibility $\chi_e$ is given by the following equations.
\[\varepsilon = \varepsilon_0 + \frac{Nq^2}{m}\frac{1}{\left(-\omega^2 - i\gamma\omega +\omega_0^2\right)}\]\[\varepsilon_r = 1 + \frac{Nq^2}{\varepsilon_0m}\frac{1}{\left(-\omega^2 - i\gamma\omega +\omega_0^2\right)}\]\[\chi = \frac{Nq^2}{\varepsilon_0m}\frac{1}{\left(-\omega^2 - i\gamma\omega +\omega_0^2\right)}\]

Definition 2.4.4  The Drude model is a special case of the Lorentz oscillator, when $\omega_0=0$.

Definition 2.4.5  A 2nd order nonlinear oscillator is a model for the polarization of bound electrons in a solid based on a charged mass $m$ with charge $q$, density $N$, a quadratic restoring force and a damping coefficient $\gamma$. The position of the charge in the potential well is $x(t)$.
\[\frac{d^2x}{dt^2} + \gamma\frac{dx}{dt} + \omega_0^2x + ax^2 = \frac{q}{m}E(t)\]\[P(t) = Nqx(t)\]

Definition 2.4.6  A material has dispersion if the index of refraction changes at different frequencies.

Definition 2.4.7  A material has normal dispersion in regions where the index of refraction $n$ increases as frequency $f$ increases, that is $\frac{dn}{df}>0$.

Definition 2.4.8  A material has anomalous dispersion in regions where the index of refraction $n$ decreases as frequency $f$ increases, that is $\frac{dn}{df}<0$.

Definition 2.4.9  The process of photo-bleaching is when existing resonance in a material is suppressed due to photo excitation.

2.5 Phase MatchingN7T2

Definition 2.5.1  The wave vector denoted $\vec{k}$ of a plane wave is the vector with magnitude $k = \frac{2\pi}{\lambda}$ that points in the direction of the wave front such that $e^{-(\vec{k}\cdot\vec{r}-\omega t)}$ describes a wave of angular frequency $\omega$ at position $\vec{r}$ and time $t$.

Corollary 2.5.2  The magnitude of the wave vector $\vec{k}$ of a plane wave propagating in media with wavelength $\lambda$, frequency $f$, angular frequency $\omega$, velocity $v$ and index of refraction $n$ can be written as any of the following expressions, where $c$ is the speed of light.
\[k = \frac{2\pi}{\lambda} = \frac{2\pi f}{v} = \frac{2\pi n f}{c} = \frac{n\omega}{c} \]

Definition 2.5.3  The d-effective constant denoted $d_{eff}$ is half of the effective $\chi_e^{(2)}$ coefficient of a material in the direct of the electric field used for a $\chi_e^{(2)}$ nonlinear process.
\[d_{eff} = \frac{\chi_e^{(2)}}{2}\]

Result 2.5.4  The 1D wave equation in a material can be derived by plugging the 1D nonlinear susceptibility into the wave equation from polarization. It describes the motion of light through a material with a source term generated from the nonlinear component of the polarization.
\[\frac{\partial^2E}{\partial t^2} - \mu_0\varepsilon\frac{\partial^2{E}}{\partial t^2}=\mu_0\frac{\partial^2P^{(NL)}}{\partial t^2}\]

Result 2.5.5  The differential equation for 1D sum frequency generation is the resulting differential equation describing sum frequency generation where two plane waves with angular frequencies $\omega_1<\omega_2$ generate a sum frequency wave with angular frequency $\omega_3 = \omega_1+\omega_2$ derived by plugging in polarization $P^{(NL)}(z,t) = 4\varepsilon_0d_{eff}E_1E_2e^{\pm i((k_1+k_2)z-\omega_3t)}$ and electric field $E(z,t) = E_3(z)e^{\pm i(k_3z-\omega_3t)}$ into the 1D wave equation in a material.
\[\frac{\partial^2 E_3(z)}{\partial z^2} \pm 2ik_3 \frac{\partial E_3(z)}{\partial z} = \frac{-4d_{eff}\omega_3^2}{c^2}E_1E_2e^{\pm i(k_1+k_2 - k_3)z}\]

Definition 2.5.6  The slowly varying amplitude approximation assumes that the fractional change of the amplitude of one of the waves in a nonlinear process on the distance of the wavelength $\lambda$ is small.$\newcommand\abs[1]{\left|#1\right|}$
\[\abs{\frac{\partial^2 E}{dz^2}} << \abs{k\frac{\partial E}{\partial z}}\]

Result 2.5.7  If the slowly varying amplitude approximation is made for the differential equation for 1D sum frequency generation, then the following differential equations follows.
\[\frac{\partial E_3(z)}{\partial z} =\frac{\pm 2id_{eff}\omega_3^2}{k_3c^2}E_1E_2e^{\pm i(k_1+k_2 - k_3)z}\]

Definition 2.5.8  The phase matching coefficient denoted $\Delta k$ is defined $\Delta k = k_1+k_2-k_3$.

Corollary 2.5.9  The phase matching condition for 1D sum frequency generation is when $k_1 + k_2 = k_3$ or when $\Delta k = 0$. When this condition is met, high intensity $\omega_3$ field can be generated.

Result 2.5.10  The peak amplitude of 1D sum frequency generation $E_3(z)$ at position $z$ in the crystal is given by the following equation when the slowly varying amplitude approximation is applied.
\[E_3(z) = \frac{\pm 2id_{eff} \omega_3^2 E_1E_2}{k_3 c^2}\int_0^{z}e^{i(k_1+k_2-k_3)\ell}d\ell = \frac{\pm 2id_{eff} \omega_3^2 E_1E_2}{k_3 c^2}\left(\frac{e^{i(k_1+k_2-k_3)z} - 1}{i(k_1+k_2-k_3)}\right)\]

Corollary 2.5.11  The peak intensity of 1D sum frequency generation $I_3(z)$ at position $z$ in the crystal is given by the following equation when the slowly varying amplitude approximation is applied.$\newcommand\abs[1]{\left|#1\right|}$
\[I_3(z) = 2n_3\varepsilon_0c\abs{E_3}^2 = \frac{8n_3\varepsilon_0d_{eff}^2 \omega_3^4 \abs{E_1}^2\abs{E_2}^2}{k_3^2 c^3}\abs{\frac{e^{i\Delta k z}-1}{\Delta k}}^2\]\[I_3(z) = \frac{8d_{eff}^2 \omega_3^4 \abs{E_1}^2\abs{E_2}^2}{n_1n_2n_3\varepsilon_0c^2}z^2 \text{sinc}^2\left(\frac{\Delta kz}{2}\right)\]

2.6 BirefringenceCA04

Definition 2.6.1  An anisotropic linear dielectric is a linear dielectric where the displace field $\vec{D}$ is related to the elecctric field $\vec{E}$ by a non-trivial linear permittivity matrix $\mathcal{E}$.
\[ \vec{D} = \mathcal{E}\vec{E}\]

File 2.6.2  Birefringence.pdf

2.7 Critical Phase MatchingJ7C2

Definition 2.7.1  The process of critical phase matching is when birefringence is used to achieve phase matching.

Definition 2.7.2  The ordinary ray is the ray of light that does not experience extraordinary refraction.

Definition 2.7.3  The extraordinary ray is the ray of light that does experience extraordinary refraction.

Result 2.7.4  For a birefringent crystal with extraordinary index of refraction $\bar{n}_e$ and ordinary index of refraction $n_o$, the following equation can be used to calculate the index of refraction that a ray of light with polarization at angle $\theta$ from the ordinary axis of a uniaxial crystal.
\[\frac{1}{(n(\theta))^2} = \frac{\sin^2\theta}{\bar{n}_e^2} + \frac{\cos^2\theta}{n_o}\]

Definition 2.7.5  Type I phase matching TODO

Definition 2.7.6  Type II phase matching TODO

2.8 Alternate Phase MatchingW2T2

Definition 2.8.1  The quasi-phase matching technique is the construction of a crystal where the crystal is periodically inverted to flip the sign of $\chi_e^{(2)}$ when the intensity of the desired nonlinear process is just about to decrease. This allows for significant nonlinear generation without phase matching.

Definition 2.8.2  The temperature tuning or the non-critical phase matching technique is where temperature dependent birefringent crystals are heated or cooled to achieve phase matching.

Definition 2.8.3  The non-collinear phase matching technique is the use of non-collinear beam to satisfy the phase matching condition. The phase matching condition $\vec{k}_1+\vec{k}_2 = \vec{k_3}$ can be satisfied by sending in the two generating beams at different opposing angles such that when added together they produce $\vec{k}_3$.

Definition 2.8.4  The 4-wave mixing technique is a $\chi_e^{(3)}$ process where 3 beams are combined to generate a fourth beam.

Definition 2.8.5  The tilted pulse front generation technique is $\chi_e^{(2)}$ nonlinear generation at the Cherencov angle as the pulse passes through a nonlinear material. This generation can be phase matched by sending in a tilted pulse front at that angle such that the generated light constructively interferes.

2.9 Operational Parametric AmplifiersC53C

Definition 2.9.1  A parametric nonlinear process is a process where the initial and final energy states of the material is the same.

Definition 2.9.2  A non-parametric nonlinear process is a process where the energy states of the material are affected by the process.
nobel prize 2003 and nobel prize 2005

2.10 Ramen ScatteringJD0A

Definition 2.10.1  A two photon absorption process is is any process where two photons are absorbed simultaneously to accesses an otherwise inaccessible energy state. These processes typically require extremely high intensity for there to be enough photons overlapping for the effect to be significant.

Definition 2.10.2  A stimulated Ramen scattering experiment is where a narrow band pump is sent into a sample and the resulting emission spectra is measured. This can be used to reveal the energy levels of the state available for excitation.

Definition 2.10.3  A Stokes scattering process refers to scattered light produced with a lower energy than the pump. This occurs when a phonon is emitted into the material after the initial excitation.

Definition 2.10.4  An anti-Stokes scattering process refers to scattered light produced with a higher energy than the pump. This occurs when a phonon is absorbed from the material after the initial excitation.

Image 2.10.5  Types of Ramen Scattering 

Definition 2.10.6  A saturable absorber is a material that will become "saturated" at high intensity and won't be able to absorb as much light. The relationship between the absorption rate $\alpha$ and intensity $I$ is described by the following equations.
\[\alpha = \frac{\alpha_0}{1 + I/I_0},\quad I(z) = I_0e^{-\alpha z}\]
where $\alpha_0$ is the linear absorption at low intensity, $I_0$ is the saturation intensity and $z$ is the distance through the crystal.

Definition 2.10.7  The Kerr effect is a process where non-zero $\chi_e^{(3)}$ leads to an index of refraction that changes with intensity.

Definition 2.10.8  The self focusing process is a Kerr effect driven process where a positive $\chi_e^{(3)}$ causes a high intensity light pulse to be focused by it's own induced refractive index in a material.

Definition 2.10.9  A Kerr lens is a apparent lens produced by a self focusing at high intensity.

3 Ultrafast Lasers8CMT

3.1 Keys to Laser OperationJK38

Definition 3.1.1  The stimulated absorption process is when a photon is absorbed into a quantum state in a material.

Definition 3.1.2  The stimulated emission process is when a photon interacts with a quantum state and causes another photon to be emitted from the decay of that quantum state. The emitted photon will have the same frequency and phase as the stimulating photon.

Definition 3.1.3  The spontaneous emission process is when a phonon is spontaneously emitted from the decay of a quantum state. This process is mediated by virtual photons produced by the vacuum.

Definition 3.1.4  A *two-level system is a quantum system with two energy levels that particles can occupy.

Result 3.1.5  The rate equation for a two-level system describes how the density of particles in the excited state changes over time. Let $N_1$ and $N_2$ be the densities of ground and excited states respectively, $\rho$ be the incident photon energy density,$B_{1\to2},B_{2\to1}$ and $A_{21}$ be the Einstein coupling coefficients for stimulated absorption, stimulated emission and spontaneous emission respectively.
\[\frac{dN_2}{dt}=B_{1\to 2}\rho N_1 - B_{2\to1}\rho N_2 - A_{2\to1} N_2 \]

3.2 Laser CavitiesAH4A

Definition 3.2.1  A ray vector is a 2d vector representing a ray of light, where the first component represents the displacement from the center and the second represents the angle.

Definition 3.2.2  An ABCD matrix is a 2 by 2 matrix used to perform approximate ray tracing calculations for lenses and mirrors. An ABCD matrix acts on a ray vector via matrix multiplication. For small angles (\sin\theta \approx \theta) this is a valid approximation.

3.2.3 

3.3 

Statistical MechanicsRPCH

$\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}$

1 Thermodynamics68RC

1.1 1st Law of ThermodynamicsW8N0

Definition 1.1.1  Temperature denoted $T$ is the macroscopically measurable state function that is only equal when two systems are in thermodynamic equilibrium.

Definition 1.1.2  Work denoted $W$ is energy transferred to a system by macroscopic forces.

Definition 1.1.3  Heat denoted $Q$ is energy transferred to a system by microscopic forces.

Definition 1.1.4  Quasi-static processes are slow transformations where the macroscopic properties remain well defined.

Definition 1.1.5  Diathermic walls are walls that allow heat transfer.

Definition 1.1.6  Adiabatic walls are walls that don't allow heat transfer.

Law 1.1.7  The 1st Law of Thermodynamics states that the exact differential energy $dE$ of a system is the sum of the inexact differential heat into the system $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj Q$ and the inexact differential work done on the system $\dj W$.
\[dE = \dj Q + \dj W\]

Definition 1.1.8  Heat engine efficiency $\eta = \frac{W}{Q_H} = \frac{Q_H-Q_C}{Q_H}$ is the amount of work extracted per unit heat consumed.

Definition 1.1.9  Refrigerator coefficient of performance $\omega = \frac{Q_C}{W} = \frac{Q_C}{Q_H-Q_C}$ is the amount of heat extracted per unit work used.

Result 1.1.10  The Carnot engine $\eta_{CE} = \frac{T_H-T_C}{T_H} = 1-\frac{T_C}{T_H}$ is the most efficient possible heat engine.

Result 1.1.11  The Carnot refrigerator $\omega_{CR} = \frac{T_H-T_C}{T_C} = \frac{T_H}{T_C}$ is the most efficient possible refrigerator.

1.2 2nd Law of ThermodynamicsKEWA

Definition 1.2.1  The entropy of a system is the state function $S$ such that the following holds:
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}dS = \frac{dE}{T} + \frac{\mathbf{P}\cdot d\mathbf{V}}{T}\]
where $\mathbf{P}$ are the generalized pressures of the system and $\mathbf{V}$ are the generalized volumes of the system.

Theorem 1.2.2  Clausius's Theorem states that for an arbitrary cyclic process in phase space $\mathbf{\lambda}$, the integral of differential heat over temperature is non-positive.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)} d\mathbf{\lambda}\leq 0\]

Law 1.2.3  The 2nd Law of Thermodynamics states that the entropy of a closed system is non-decreasing.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS \geq \frac{\dj Q}{T}\]

Definition 1.2.4  Reversible processes are processes where $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS = \frac{\dj Q}{T}$ during the entire process.

Definition 1.2.5  Irreversible processes are processes where $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}} dS > \frac{\dj Q}{T}$ at some point during the process.

Result 1.2.6  Reversible processes are quasi-static, path independent and do not change the entropy of the system.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)}\partial\mathbf{\lambda} = S_f - S_i = 0\quad\text{ (for reversible processes)}\]

Result 1.2.7  Irreversible processes result in a net increase in entropy.
\[\oint\frac{\dj Q(\mathbf{\lambda})}{T(\lambda)}\partial\mathbf{\lambda} > S_f - S_i > 0\quad\text{ (for irreversible processes)}\]

1.3 Open and Closed SystemsDMK9

Definition 1.3.1  The partial pressures $\{P_i\}$ are the pressures of the system exerted by each type of particle in the system.

Definition 1.3.2  The volumes $\{V_i\}$ are the amounts of space occupied by each type of particle in the system.

Definition 1.3.3  The generalized forces $\{J_i\}$ are the forces acting on the system.

Definition 1.3.4  The generalized displacements $\{x_i\}$ are the displacements of the system for each of the generalized forces acting on the system.

Definition 1.3.5  The physical work $W_{phy}$ is the work done on the system by generalized forces or partial pressures of the system.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W_{phy} = \sum_i{J_i dx_i} = \mathbf{J}\cdot d\mathbf{x} = \sum_i{-P_i dV} = -\mathbf{P}\cdot d\mathbf{V}\]

Definition 1.3.6  The chemical work $W_{chem}$ is the work done on the system by the generalized chemical potentials $\{\mu_\alpha\}$ and the generalized numbers of particles $\{N_\alpha\}$ which is the work done by a change in particle number.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W_{chem} = \sum_\alpha{\mu_\alpha dN_\alpha} = \mathbf{\mu}\cdot d\mathbf{N}\]

Definition 1.3.7  A closed system is a system that cannot exchange particles with the environment, that is $d\mathbf{N} = 0$.

Definition 1.3.8  An open system is a system that can exchange particles with the environment.

Result 1.3.9  The total work $W$ done on a system is the sum of the physical work $W_{phy}$ and the chemical work $W_{chem}$.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\dj W = \dj W_{phy},\quad \quad \text{Closed System}\]\[\dj W = \dj W_{phy} + \dj W_{chem},\quad \text{Open System}\]

Result 1.3.10  Differential energy states that the following holds for all thermodynamic systems.
\[\partial E = T\partial S - \mathbf{P}\cdot\partial\mathbf{V} + \mathbf{\mu}\cdot d\mathbf{N}\]\[T = \left( \frac{\partial E}{\partial S} \right)_{\mathbf{V},\mathbf{N}},\quad -P_i=\left( \frac{\partial E}{\partial V_i} \right)_{S,V_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial E}{\partial N_\alpha} \right)_{S,\mathbf{V},N_{\beta\neq \alpha}}\]

Result 1.3.11  Differential entropy states that the following holds for all thermodynamic systems.
\[\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}\partial S = \frac{\partial E}{T} + \frac{\mathbf{P}\cdot\partial\mathbf{V}}{T} - \frac{\mathbf{\mu}\cdot d\mathbf{N}}{T}\]\[\frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{\mathbf{V},\mathbf{N}},\quad \frac{P_i}{T} = \left( \frac{\partial S}{\partial V_i} \right)_{E,V_{j\neq i},\mathbf{N}},\quad \frac{\mu_\alpha}{T}=\left( \frac{\partial S}{\partial N_\alpha} \right)_{E,\mathbf{V},N_{\beta\neq \alpha}}\]

Result 1.3.12  The Gibbs-Duhem Relation states that $T$, $\mathbf{P}$ and $\mathbf{\mu}$ are related by the following differential equation.
\[SdT - \mathbf{V}\cdot d\mathbf{P} + \mathbf{N}\cdot d\mathbf{\mu} = 0\]

1.4 Enthalpy42CR

1.4.1  The enthalpy is a state function $H$ defined by
\[H = E+\mathbf{P}\cdot\mathbf{V}\]

Result 1.4.2  For adiabatic systems with constant external forces, the enthalpy is minimized.
\[dH\leq 0\]

Result 1.4.3  Differential enthalpy states that the following holds for all thermodynamic systems.
\[dH = TdS + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\]\[T = \left( \frac{\partial H}{\partial S} \right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial H}{\partial P_i}\right)_{S,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial H}{\partial N_\alpha} \right)_{S,\mathbf{P},N_{\beta\neq \alpha}}\]

1.5 Helmholtz Free Energy9A0J

Definition 1.5.1  Isothermal processes are processes where the temperature is constant.

Definition 1.5.2  The Helmholtz free energy is a state function $F$ defined by
\[F = E-TS\]

Result 1.5.3  For isothermal systems with no external work, the Helmholtz free energy is minimized.
\[dF \leq 0\]

Result 1.5.4  Differential Helmholtz free energy states that the following holds for all thermodynamic systems.
\[dF = -SdT - \mathbf{P}\cdot d\mathbf{V} + \mathbf{\mu}\cdot d\mathbf{N}\]\[-S = \left( \frac{\partial F}{\partial T} \right)_{\mathbf{V},\mathbf{N}},\quad -P_i = \left(\frac{\partial F}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial F}{\partial N_\alpha} \right)_{T,\mathbf{V},N_{\beta\neq \alpha}}\]

1.6 Gibbs Free Energy2MJ8

Definition 1.6.1  The Gibbs free energy is a state function $G$ defined by
\[G = E-TS+\mathbf{P}\cdot\mathbf{V}\]

Result 1.6.2  For isothermal systems with constant external forces, Gibbs free energy is minimized.
\[dG\leq 0\]

Result 1.6.3  Differential Gibbs free energy states that the following holds for all closed thermodynamic systems.
\[dG =-SdT + \mathbf{V}\cdot d\mathbf{P} + \mathbf{\mu}\cdot d\mathbf{N}\]\[-S = \left(\frac{\partial G}{\partial T}\right)_{\mathbf{P},\mathbf{N}},\quad V_i = \left(\frac{\partial G}{\partial P_i}\right)_{T,P_{j\neq i},\mathbf{N}},\quad \mu_\alpha=\left( \frac{\partial G}{\partial N_\alpha} \right)_{T,\mathbf{P},N_{\beta\neq \alpha}}\]

1.7 Grand Potential52CC

Definition 1.7.1  The grand potential is a state function $\mathcal{G}$ defined by
\[\mathcal{G} = E - TS - \mathbf{\mu}\cdot\mathbf{N}\]

Result 1.7.2  For isothermal systems in chemical equilibrium with no external work, the grand potential is minimized
\[d\mathcal{G}\leq 0\]

Result 1.7.3  Differential grand potential states that the following holds for all thermodynamic systems.
\[d\mathcal{G} = -SdT - \mathbf{P}\cdot d\mathbf{V} - \mathbf{N}\cdot d\mathbf{\mu}\]\[-S = \left(\frac{\partial \mathcal{G}}{\partial T}\right)_{\mathbf{V},\mathbf{\mu}},\quad -P_i = \left(\frac{\partial \mathcal{G}}{\partial V_i}\right)_{T,V_{j\neq i},\mathbf{\mu}},\quad -N_\alpha=\left( \frac{\partial \mathcal{G}}{\partial \mu_\alpha} \right)_{T,\mathbf{V},\mu_{\beta\neq \alpha}}\]

1.8 3rd Law of ThermodynamicsN6R8

Law 1.8.1  The 3rd law of thermodynamics states that the limit of entropy as temperature approaches to zero is a universal constant.
\[\lim_{T\to 0}{S(T,\dots)} = C\]

2 Fundamental Statistical MechanicsC676

2.1 Microcanonical Ensemble8TMK

Definition 2.1.1  The multiplicity function $\Omega$ of a system is the number of possible microstates for a given macrostate.

Law 2.1.2  The Boltzmann Hypothesis states that probability of all possible microstates are equal for a particular macrostate
\[\mathscr{p}_i = \frac{1}{\Omega}\]

Definition 2.1.3  The microcanonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[\mathscr{p}_i = \frac{1}{\Omega(E,\mathbf{V},\mathbf{N})}\]

Definition 2.1.4  The microcanonical entropy $S$ of a system is the Boltzmann constant times the natural log of the multiplicity function.
\[S = k_B\log\Omega\]

Definition 2.1.5  The temperature $T$ and thermodynamic temperature $\beta$ of a system are defined in terms of the derivative of energy with respect to entropy.
\[T = \frac{\partial E}{\partial S} = \frac{\partial E}{\partial k_B \log\Omega} = \frac{1}{k_B\beta}\]\[\beta = \frac{\partial}{\partial E}\log\Omega = \frac{1}{k_B T}\]

Proposition 2.1.6  Stirling's Approximation states that for sufficiently large $N$, the natural log of $N!$ can be approximated.
\[\log(N!) \approx N\log(N) - N\]

Definition 2.1.7  The ensemble average denoted $\langle \mathscr{O}\rangle$ of a variable $\mathscr{O}$ is sum of the value for all microstates weighted by their probabilities.
\[\langle \mathscr{O}\rangle = \sum_{i}{\mathscr{O}_i\mathscr{p}_i}\]

2.2 Canonical EnsemblePNJT

Definition 2.2.1  The canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$ and particles numbers $\mathbf{N}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the thermodynamic temperature $\beta$ and the partition function $z$.
\[\mathscr{p}_i = \frac{1}{\Omega(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\]

Definition 2.2.2  The canonical energy E of a system in the canonical ensemble is the ensemble average of energy.
\[\langle E\rangle = \sum_{i}{E_i \mathscr{p}_i} = \frac{1}{z}\sum_{i}{\frac{-\partial}{\partial \beta}e^{-\beta E_i}} = -\frac{1}{z}\frac{\partial z}{\partial \beta} = -\frac{\partial}{\partial \beta}\log z\]

Definition 2.2.3  The heat capacity of a system $C_V$ is the derivative of canonical energy in terms of temperature.
\[C_V = \left(\frac{\partial E}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}\log z = -k_B \beta^2 \frac{\partial^2}{\partial\beta^2}(\beta F) = k_B\beta^2(\langle E^2\rangle - \langle E\rangle^2)\]

Result 2.2.4  The Helmholtz free energy $F$ can be written in terms of the temperature and the partition function.
\[F = -\frac{1}{\beta}\log z\]

Result 2.2.5  The pressure $P$ of a system can be written as the ensemble average of pressure for each microstate.
\[P = -\left(\frac{\partial F}{\partial V}\right)_{T,\mathbf{N}} = \frac{\partial}{\partial V}\left(\frac{1}{\beta}\log z\right) = \frac{1}{z}\sum_i{\left(\frac{-\partial E}{\partial V}\right)e^{-\beta E_i}} = \langle P_i\rangle\]

Result 2.2.6  The entropy $S$ of a system can be written in terms of the ensemble average of log of the probability of each microstate.
\[S = -\left(\frac{\partial F}{\partial T}\right)_{\mathbf{V},\mathbf{N}} = k_B\beta^2\left(\frac{\partial F}{\partial \beta}\right)_{\mathbf{V},\mathbf{N}} = -k_B\beta^2\left(\frac{\partial}{\partial \beta}\frac{1}{\beta}\log z\right)_{\mathbf{V},\mathbf{N}} = \frac{k_B}{z}\sum_i{e^{-\beta E_i}(\log z +\beta E_i)}\]

Proposition 2.2.7  Geometric series convergence states that for $|r|<1$ the following infinite series converges to $1/(1-r)$.
\[\sum_{k=0}^\infty{r^k} = \frac{1}{1-r}\]

2.3 Ideal Gas in the Canonical Ensemble99EH

Definition 2.3.1  A free particle in the canonical ensemble is a system of a single quantum mechanical particle is a cubic box of volume $V=L^3$.
\[\varepsilon_{\vec{n}} = \frac{\hbar^2\pi^2}{2mL}\left(n_x^2 + n_y^2+ n_z^2\right)\]\[z = \sum_{\vec{n}}e^{-\beta\varepsilon_\vec{n}} = \frac{V}{\ell_Q^3}\]

Definition 2.3.2  The Debroglie thermal wavelength denoted $\ell_Q$ is the average wavelength of particles in a free particles system or an ideal gas.
\[\ell_Q = \sqrt{\frac{2\pi\hbar^2}{mk_BT}}\]

Result 2.3.3  The total partition function of non-interacting systems is the product of their individual partition functions.
\[z_{\text{tot}} = z_1z_2\dots\]

Definition 2.3.4  An ideal gas in the canonical ensemble is a system of $N$ indistinguishable non-interacting free particles with the Gibbs factor $\frac{1}{N!}$ to account for the indistinguishably of the quantum particles.
\[z = \frac{1}{N!}z_1^N = \frac{1}{N!}\left(\frac{V}{\ell_Q^3}\right)^N\]

Result 2.3.5  The Helmholtz free energy of an ideal gas with the Gibbs factor in the canonical ensemble is
\[F = k_BT\log\frac{\ell_Q^3}{V} + k_BTN\log N - k_BTN\]

Result 2.3.6  The ideal gas law states that for an ideal gas in the canonical ensemble,
\[PV=Nk_BT\]

Result 2.3.7  The average energy of an ideal gas in the canonical ensemble is
\[\langle E\rangle = \frac{3}{2}Nk_BT\]

Result 2.3.8  The entropy of an ideal gas with the Gibbs factor in the canonical ensemble is
\[S = -\left(\frac{\partial F}{\partial T}\right)_{N,V} = k_BN\left[\frac{5}{2}-\log\frac{N\ell_Q^3}{V}\right]\]

2.4 Grand Canonical Ensemble7Z98

Definition 2.4.1  The grand-canonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$. The probability of a particular microstate $i$ is written in terms of the energy of the microstate $E_i$, the particle numbers of the microstate $\mathbf{N}_i$, the thermodynamic temperature $\beta$, the chemical potentials $\mathbf{\mu}$ and the grand partition function $z$.
\[\mathscr{p}_i = \frac{1}{\Omega(\mathbf{T},\mathbf{V},\mathbf{\mu})}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}\]

Result 2.4.2  The grand potential $\mathcal{G}$ can be written in terms of the temperature and the grand partition function.
\[\mathcal{G} = -\frac{1}{\beta}\log \mathscr{z}\]

2.5 Classical Statistical MechanicsARJH

Definition 2.5.1  The Hamiltonian denoted $\mathcal{H}$ of a classical system is a function that represents the total energy of the system.

Law 2.5.2  Hamilton's Equations state that classical systems with Hamiltonian $\mathcal{H}$ evolve according to the following differential equations, where $q$ is the position and $p$ is the momentum.
\[\frac{\partial q_i}{\partial t}=\frac{\partial \mathcal{H}}{\partial p_i},\quad \frac{\partial p_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial q_i}\]

Definition 2.5.3  The probability density function denoted $\rho(q,p)$ is the function whose integral represents the probability of finding a classical system in a given region of phase space.

Theorem 2.5.4  The Liouville Theorem states that for classical systems with Hamiltonian $\mathcal{H}$ and probability density $\rho$,
\[\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho,\mathcal{H}\},\]
where $\{\ ,\ \}$ is a Poisson bracket.

Definition 2.5.5  The classical microcanonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[\rho(p,q) =\frac{\delta(E-\mathcal{H})}{(2\pi\hbar)^{3N}}\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\delta(E-\mathcal{H})\]

Result 2.5.6  For large $N$, the following multiplicity functions are equivalent.
\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\delta(E-\mathcal{H})\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}(\Theta(E-\mathcal{H})-\Theta(E-\Delta-\mathcal{H}))\]\[\Omega(E) = \int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}\Theta(E-\mathcal{H})\]

Definition 2.5.7  A classical ideal gas in the microcanonical ensemble is the system with hamiltonian $\mathcal{H}$ defined by
\[\mathcal{H} = \sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}\]\[\Omega \approx \frac{2\pi^{3N/2}}{(3N/2 - 1)!} \frac{2m^{3N/2}}{N!(2\pi\hbar)^{3N}3N} \frac{3N\Delta}{2E}V^NE^{3N/2}\]

Definition 2.5.8  The classical canonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and particles numbers $\mathbf{N}$.
\[\rho(\mathbf{p},\mathbf{q}) = \frac{e^{-\beta \mathcal{H}}}{z}\]\[z = \frac{1}{N!}\int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}e^{-\beta\mathcal{H}}\]\[z = \frac{1}{N!}(z_1)^N,\quad z_1 = \int\frac{d^{3}qd^{3}p}{(2\pi\hbar)^{3}}e^{-\beta\mathcal{H}}\]

Definition 2.5.9  The classical grand-canonical ensemble is the ensemble of statistical mechanics for classical systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$.
\[\rho(\mathbf{p},\mathbf{q},\mathbf{N}) = \frac{e^{-\beta (\mathcal{H}_\mathbf{N}-\mathbf{\mu}N)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_N\frac{1}{N!}\int\frac{d^{3N}qd^{3N}p}{(2\pi\hbar)^{3N}}e^{-\beta(\mathcal{H}_\mathbf{N}-\mathbf{\mu}\mathbf{N})}\]

Theorem 2.5.10  The equipartition theorem states that the average energy is proportional to temperature times the number of nonzero coefficients for a Hamiltonian of the following form.
\[\mathcal{H} = \sum_jA_jp_j^2 + \sum_jB_jq_j^2\]\[\langle\mathcal{H}\rangle = \frac{1}{2}fk_BT\]
where $f$ is the number of non-vanishing coefficients $A_j$ and $B_j$.

2.6 Quantum Statistical Mechanics05AR

Definition 2.6.1  The density operator denoted $\hat{\rho}$ is the quantum mechanical operator that represents the probability of a state.
\[\hat{\rho} = \sum_\alpha\mathscr{p}_\alpha\ket{\Psi_\alpha(t)}\bra{\Psi_\alpha(t)}\]
where $\mathscr{p}_\alpha$ is the probability of the state $\ket{\Psi_\alpha(t)}$.

Result 2.6.2  The trace of the density operator is one, $\text{Tr}(\hat{\rho}) = \sum_\alpha\mathscr{p}_\alpha = 1$.

Result 2.6.3  The density operator is Hermitian with eigenstates $\ket{\rho_i}$ and corresponding real eigenvalues $w_i$, such that
\[\hat{\rho} = \sum_iw_i\ket{\rho_i}\bra{\rho_i}.\]

Result 2.6.4  The square of the density operator is itself, if and only if the system is a pure quantum state $\ket{\Psi}$. \[\hat{\rho}^2=\hat{\rho} \quad \Leftrightarrow \quad \hat{\rho} = \ket{\Psi}\bra{\Psi}\]

Result 2.6.5  The time dependence of the density operator is given by the commutation relation of the Hamiltonian with the density operator.
\[i\hbar\frac{\partial }{\partial t}\hat{\rho}(t) = [\hat{H},\hat{\rho}(t)]\]

Result 2.6.6  For systems in equilibrium, there exist simultaneous eigenvalues $\ket{E_i}$ of $\hat{H}$ and $\hat\rho$ such that
\[\hat{\rho}\ket{E_i} = w_i\ket{E_i},\quad \hat{H}\ket{E_i} = E_i\ket{E_i}.\]\[\hat{\rho} = \sum_iw_i\ket{E_i}\bra{E_i}.\]

Definition 2.6.7  The quantum ensemble average denoted $\langle \mathscr{O}\rangle$ of a variable $\mathscr{O}$ is sum of the quantum expectation value for all microstates weighted by their probabilities.
\[\langle \mathscr{O}\rangle = \sum_{\alpha}{\mathscr{p}_\alpha \bra{\Psi_\alpha(t)} \hat{\mathscr{O}} \ket{\Psi_\alpha(t)} } = \text{Tr}(\hat{\rho}\hat{\mathscr{O}}) = \sum_n{\bra{E_n}\hat{\rho}\hat{\mathscr{O}}\ket{E_n}} = \sum_n{w_n\bra{E_n}\hat{\mathscr{O}}\ket{E_n}}\]

Definition 2.6.8  The quantum microcanonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by the energy $E$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$.
\[w_i = \frac{\delta_{E,E_i}}{\Omega(E)}\]

Definition 2.6.9  The quantum canonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and particles numbers $\mathbf{N}$.
\[w_i = \frac{e^{-\beta E_i}}{\text{Tr}(e^{-\beta \hat{H}})} = \frac{e^{-\beta E_i}}{\sum_{j}\bra{E_j}e^{-\beta \hat{H}}\ket{E_j}} = \frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \text{Tr}(e^{-\beta \hat{H}}) = \sum_{j}\bra{E_j}e^{-\beta \hat{H}}\ket{E_j} = \sum_{j}{e^{-\beta E_j}}\]

Definition 2.6.10  The quantum grand-canonical ensemble is the ensemble of statistical mechanics for quantum systems where the macrostates are described by temperature $T$, volumes $\mathbf{V}$, and chemical potentials $\mathbf{\mu}$.
\[w_{i,N} = \frac{e^{-\beta (E_{i,N}-\mu{N})}}{\text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) })} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\sum_N\sum_{j}e^{-\beta (E_j-\mu N)}} = \frac{e^{-\beta (E_{i,N}-\mu N)}}{\mathscr{z}}\]\[\mathscr{z} = \text{Tr}_{FS}(e^{-\beta (\hat{H}-\mu\hat{N}) }) = \sum_N\sum_{j}e^{-\beta (E_j-\mu N)}\]

3 Quantum GasesPT46

3.1 Identical ParticlesR6AK

Definition 3.1.1  The exchange operator is the operator $P_{a,b}$ that exchanges the quantum states of particles $a,b$.
\[P_{a,b}\Psi(a,b) = \Psi(b,a)\]

Result 3.1.2  For identical particles, the exchange operator can introduce a phase factor of $\phi = 0$ or $\phi = \pi$.
\[P_{a,b}\Psi(a,b) = \Psi(b,a) = e^{i\phi}\Psi(a,b) = \eta\Psi(a,b)\quad\text{ where }\eta = \pm 1\]

Definition 3.1.3  The permutation operator is the exchange operator generalized to $N$ particle systems that performs a permutation $P$ on the quantum states of particles in the system.
\[\hat{P}\Psi(1,2,\dots, N) = \Psi(P(1),P(2),\dots,P(N))\]

Definition 3.1.4  The parity of a permutation denoted $\sigma(P)$ is the minimum number of pairwise swaps of the permutation $P$.

Result 3.1.5  For identical particles, the permutation operator can introduce a phase factor of $\phi=0$ or $\phi = \pi$ for each pairwise swap.
\[\hat{P}\Psi(1,2,\dots, N) = \Psi(P(1),P(2),\dots,P(N)) = \eta^{\sigma(P)} \Psi(1,2,\dots,N)\quad\text{ where }\eta = \pm 1\]

Definition 3.1.6  A fermion is a particles where a sign flip is introduced by the exchange operator, that is $\eta = -1$.

Definition 3.1.7  A boson is a particle where no sign flip is introduced by the exchange operator, that is $\eta = +1$.

Definition 3.1.8  The antisymmetrizer operator denoted $\mathcal{A}$ is the operator that creates an antisymmetric quantum state.
\[\ket{k_1,k_2,\dots,k_N}_{-} = \mathcal{A}\ket{k_1,k_2,\dots,k_N} = \frac{1}{N!}\sum_{P}{(-1)^{\sigma(P)}\hat{P}\ket{k_1,k_2,\dots,k_N}}\]

Result 3.1.9  The Pauli Principle states that it is impossible to construct an antisymmetric quantum states where two particles are in the same state.

Definition 3.1.10  The symmetrizer operator denoted $\mathcal{S}$ is the operator that create a symmetric quantum state.
\[\ket{k_1,k_2,\dots,k_N}_{+} = \mathcal{S}\ket{k_1,k_2,\dots,k_N} = \frac{1}{\sqrt{N!\prod_{k}{n_k!}}}\sum_{p}{\hat{P}\ket{k_1,k_2,\dots,k_N}}\]
where $n_k$ is the number of particles in state $k$.

Result 3.1.11  Generalized Quantum States for Identical Particles states that the antisymmetrized and symmetrized quantum states for identical fermions or bosons can be written with generalized notation
\[\ket{k_1,k_2,\dots,k_N}_{\eta} = \frac{1}{\sqrt{N! \prod_{k}n_k!}}\sum_{p}\eta^{\sigma(P)}\hat{P}\ket{k_1,k_2,\dots,k_N}\]\[\eta = -1 \text{ for Fermions},\quad \eta = +1 \text{ for Bosons}\]\[n_k \in \{0,1\}\text{ for Fermions},\quad n_k \in \{0,1,2,3,\dots\} \text{ for Bosons}\]\[\sum_k{n_k} = N\]

Result 3.1.12  The completeness relation for identical particles states that for identical particles the completeness relation can be written as a sum of quantum states.
\[I = \sum_{k_1\leq k_2\leq \dots \leq k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}} = \frac{1}{N! \prod_{k}n_k!}\sum_{k_1,k_2,\dots,k_N} \ket{k_1,k_2,\dots,k_N}_\eta\ \prescript{}{\eta}{\bra{k_1,k_2,\dots,k_N}}\]

3.2 Quantum Gases in the Canonical Ensemble8HP1

Definition 3.2.1  The identical free particles in a box is the quantum system of $N$ particles in a large box with the following approximate eigenstates.
\[\Psi_{k_1,\dots,k_N}(x_1,\dots,x_N) = \prod_{a=1}^N\frac{e^{ik_a\cdot x_a}}{\sqrt{V}}\]\[k_a = \frac{\pi}{L}(n_{a,x},n_{a,y},n_{a,z}),\quad n_{a,x},n_{a,y},n_{a,z} \in \{1,2,\dots\}\]

Result 3.2.2  For identical free particles, the matrix element of the density operator multiplied by the partition function $\tilde{\rho}_{B,F}$ can be written in terms of the density operator for distinguishable free particles multiplied by the partition function $\tilde{\rho}_D$.
\[\text{Let } \tilde{\rho}(x_1,\dots,x_N|x_1',\dots,x_N') = z\rho(x_1,\dots,x_N|x_1',\dots,x_N') = \bra{x_1,\dots,x_N}e^{-\beta\hat{H}}\ket{x_1',\dots,x_N' }\]\[\tilde{\rho}_{B,F}(x_1,\dots,x_N|x_1',\dots,x_N') = \frac{1}{N!}\sum_P{\eta^{\sigma(P)}\tilde{\rho}_D}(x_1,\dots,x_N|x_1',\dots,x_N')\]

Result 3.2.3  The partition function for identical free particles can be written as a sum of integrals over all permutations of the $N$ particles.
\[z_{B,F} = \frac{1}{N!}\frac{1}{\ell_Q^{3N}}\sum_P{\eta^{\sigma(P)}\int{e^{\frac{-\pi}{\ell_Q^2}\sum_{a=1}^N{(x_a-x_{P(a)})^2}} dx_1,\dots,dx_N}}\]

Result 3.2.4  The partition function for identical free particles can be written as Gibbs term and the quantum exchange correction term.
\[z_{B,F} = \frac{1}{N}\left[\frac{V^N}{\ell_Q^{3N}} + \int\prod_{a=1}^N{d^3x_a}\sum_{P\neq\text{ identity}}\eta^{\sigma(P)}e^{\frac{-\pi}{\ell_Q^2}\sum_a{(x_a-x_{P(a)})^2}}\right]\]

Theorem 3.2.5  The ideal gas approximation theorem states that the ideal gas is a valid approximation when density is much larger than the square of the Debroglie thermal wavelength.
\[\ell_Q^2 >> \left(\frac{V}{N}\right)^{2/3}\]

3.3 Quantum Gases in the Grand Canonical EnsembleHNK8

Result 3.3.1  The grand partition function for identical particles can be written as a product of the grand partition functions for each single partition state, where the possible values of $n_k$ depends on whether the particles are fermions or bosons.
\[\mathscr{z} = \prod_{k}\sum_{n_k}e^{e^{-\beta(E_k-\mu)n_k}} = \prod_k{(1-\eta e^{-\beta(E_k - \mu)})^{-\eta}}\]\[\eta = \begin{cases}
-1 & \text{for fermions} \\
1 & \text{for bosons}
\end{cases}\]
where for bosons we find that $e^{-\beta(\epsilon_k - \mu)} < 1$.

Result 3.3.2  The following thermodynamic quantities can be computed for identical particles in the grand canonical ensemble.
\[\mathcal{G} = \frac{\eta}{\beta}\sum_k{\log(1-\eta e^{-\beta(E_k-\mu)})}\]\[\langle n_k \rangle = \frac{1}{e^{\beta (E_k - \mu)}-\eta}\]\[\langle E \rangle = \sum_k E_k\langle n_k \rangle = \sum_k{\frac{E_k}{e^{\beta(E_k - \mu)}-\eta}}\]\[\langle N\rangle = \sum_k \langle n_k \rangle = \sum_{k}\frac{1}{e^{\beta(E_k-\mu)}-\eta}\]

3.4 Single Particle Density of States2MA3

Definition 3.4.1  The number of accessible states denoted $\Sigma(E)$ is a function of energy that represents the number of energy states with energy less than $\epsilon$.
\[\Sigma(E) = \sum_{\alpha}\theta(E-E_\alpha)\]

Definition 3.4.2  The density of states denoted $g(E)$ is a function of energy that represents the density of states at energy $E$.
\[g(E) = \frac{\partial \Sigma(E)}{\partial E}\]

Result 3.4.3  Any sum over discrete quantum states of a function that depends on energy can be written as an energy integral of that function weighted by the density of states.
\[\sum_{\alpha}f(E_\alpha) \to \int g(E) f(E) dE\]
For identical particles the following thermodynamic quantities can be written in terms of integrals over density of states:
\[\mathcal{G} = \frac{\eta}{\beta} \int_0^\infty g(\epsilon) \log(1-\eta e^{-\beta(\epsilon-\mu)}) d\epsilon\]\[\langle E \rangle = \int_0^\infty g(\epsilon) \frac{\epsilon}{e^{\beta(\epsilon - \mu)}-\eta} d\epsilon\]\[\langle N\rangle = \int_0^\infty g(\epsilon) \frac{1}{e^{\beta(\epsilon-\mu)}-\eta} d\epsilon\]

Result 3.4.4  The density of states $g(\epsilon)$ for a spin-$S$ gas with spin degeneracy $g_S$ in a $D$-dimensional box with energy relation $\epsilon(\mathbf{p})$ is
\[g(\epsilon) = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int{d^Dp\delta(\epsilon - \epsilon(p))} = g_s\left(\frac{L}{2\pi\hbar}\right)^D \int d\Omega_D \frac{p(\epsilon)^{D-1}}{\left|\frac{\partial \epsilon}{\partial p}(p(\epsilon))\right|}\]

Definition 3.4.5  The fugacity is defined for a partitcular temperature and chemical potential as $\mathbb{z} = e^{\beta\mu}$.

3.5 Non-relativistic Fermi and Bose GasesM1FZ

Definition 3.5.1  A non-relativistic gas is a quantum gas where the energy eigenstates are related to momentum by the following relation.
\[\epsilon(p) = \frac{p^2}{2m}\]

Definition 3.5.2  The wave vector $k$ is position of the energy eigenstates in reciprocal space and is related to the momentum of the energy eigenstates.
\[k = \frac{p}{\hbar}\]\[k = \frac{\pi}{L}(n_1,n_2,\dots,n_D),\quad \text{ for }n_i\in\mathbb{N}\]\[p = \frac{\hbar\pi}{L}(n_1,n_2,\dots,n_D),\quad \text{ for }n_i\in\mathbb{N}\]

Result 3.5.3  For free particles in a box, we can convert sums of many particles into integrals of momentum or wave vectors.
\[\sum_{n_i} \to \frac{L}{2\pi\hbar}\int_{-\infty}^{\infty}{dp_i},\quad \sum_{n_i} \to \frac{L}{2\pi}\int_{-\infty}^{\infty}{dk_i}\]

Result 3.5.4  The density of states of a non-relativistic 3d Fermi and Bose gas can be derived by applying this result to $\epsilon(p) = \frac{p^2}{2m}$, $p(\epsilon) = \sqrt{2m\epsilon}$.
\[g(\epsilon) = \frac{g_sV}{\sqrt{2}\pi^2\hbar^3}m^{3/2}\sqrt{\epsilon}\]

Definition 3.5.5  The Riemann Zeta Functions for Non-relativistic Quantum Gases is the class of functions $f_m^\eta(\mathbb{z})$ of the following form.
\[f_m^\eta(\mathbb{z}) = \frac{1}{\Gamma(m)} \int_0^\infty{\frac{dx\ x^{m-1}}{\mathbb{z}^{-1}e^x -\eta}}\]

Result 3.5.6  The pressure, energy density, and density of a non-relativistic 3d Fermi and Bose gas are given by
\[\beta P = \beta \frac{\eta}{V\beta}\int_0^\infty{d\epsilon\ g(\epsilon)\log(1-\eta e^{-\beta(\epsilon-\mu)})} = \frac{g_s}{\ell_Q^3}\frac{4}{3\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{3/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{g_s}{\ell_Q^3}f_{5/2}^\eta(\mathbb{z})\]\[\beta\varepsilon = \beta \frac{E}{V} = \beta \int_0^\infty{\frac{d\epsilon\ g(\epsilon)\epsilon}{e^{\beta(\epsilon - \mu)-\eta}}}= \frac{g_s}{\ell_Q^3}\frac{2}{\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{3/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{3}{2}\frac{g_s}{\ell_Q^3}f_{5/2}^\eta(\mathbb{z})\]\[n = \frac{N}{V} = \frac{1}{V}\int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\beta(\epsilon-\mu)-\eta}}} = \frac{g_s}{\ell_Q^3}\frac{2}{\sqrt{\pi}}\int_0^\infty{\frac{dx\ x^{1/2}}{\mathbb{z}^{-1}e^x - \eta}} = \frac{g_s}{\ell_Q^3}f_{3/2}^\eta(\mathbb{z})\]

Result 3.5.7  The $f_m^\eta(\mathbb{z})$ can be expanded as a geometric series for $\mathbb{z} << 1$
\[f_m^\eta(\mathbb{z}) \approx \sum_{\alpha = 1}^\infty{\eta^{\alpha + 1}\frac{\mathbb{z}^\alpha}{\alpha^m}} = \mathbb{z} + \eta\frac{\mathbb{z}^2}{2^m} + \frac{\mathbb{z}^3}{3^m} + \cdots\]

3.6 Degenerate Fermi GasesNHWC

Definition 3.6.1  A degenerate Fermi gas is a Fermi gas at the low temperature limit $T\to 0$. In this limit the occupation function approaches a step function centered at $\epsilon_\alpha = \mu$.
\[\langle n_\alpha \rangle = f(\epsilon_\alpha) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \to \Theta(\mu(T=0) - \epsilon_\alpha)\]

Result 3.6.2  The Fermi level $\epsilon_F$ is the chemical potential of a Fermi gas at temperature goes to zero.

Proposition 3.6.3  The Fermi level can be calculated from the density of states and N by solving one of the following expressions:
\[N = g_s\sum_p\Theta(p_f-|\vec{p}|) \to g_s\left(\frac{L}{2\pi\hbar}\right)^D\int d\Omega_D\int_0^{p_F}{p^{d-1}dp}\]\[N = \int_0^{\epsilon_F}{g(\epsilon) d\epsilon}\]
The energy and degeneracy pressure can also be found with similar expressions.
\[E = g_s\sum_p\Theta(p_f-|\vec{p}|) \to g_s\left(\frac{L}{2\pi\hbar}\right)^D\int d\Omega_D\int_0^{p_F}{p^{d-1}\epsilon(p)dp}\]\[N = \int_0^{\epsilon_F}{g(\epsilon)\epsilon d\epsilon}\]

Result 3.6.4  For a degenerate non-relativistic Fermi gas the Fermi level, average energy, and degeneracy pressure are given by the following expressions.
\[\epsilon_F = \frac{\hbar^2}{2m}\left( \frac{6\pi^2 n}{g_s} \right)^{2/3}\]\[E = \frac{3}{5}N\epsilon_F\]\[P = \frac{2}{5}\frac{N}{V}\epsilon_F = \frac{2}{3}\frac{E}{V}\]

Definition 3.6.5  The Sommerfeld expansion is the expansion of the occupancy function for $T<<0$ of the following form.
\[\langle n_\alpha \rangle = f(\epsilon_\alpha) = \Theta(\mu - \epsilon) + \delta f(\epsilon)\]\[\delta f(\epsilon) = \begin{cases}f(\epsilon)=\frac{1}{e^{\beta(\epsilon - \mu)} + 1} & \epsilon > \mu \\ -f(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} & \epsilon < \mu\end{cases}\]

Result 3.6.6  The sommerfeld expansion can be applied to the chemical potential to derive the following expression.
\[\mu(T) \approx \epsilon_F(1 - \frac{\pi^2}{12}\frac{k_B^2T^2}{\epsilon_F^2})\]

3.7 Bose Einstein Condensate9N4D

Definition 3.7.1  A Bose Einstein condensate is a boson gas where a macroscopic number of particles are in the ground state for temperatures much greater than the ground state energy $k_B\tau >> \epsilon_0$.

Result 3.7.2  The number of particles in excited states $N_e$ for a boson gas can be written in terms of the density of states.
\[N_e(\tau) = \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}}\]

Theorem 3.7.3  A Bose Einstein condensate is possible when the integral approximation of $\langle N\rangle$ as $\mu \to 0$ is less than $N$.
\[N \leq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}} \quad \Rightarrow \quad \text{No BEC}\]\[N \geq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}}\quad \Rightarrow \quad \text{BEC}\]

Result 3.7.4  For a non-relativistic Bose Gas the Bose Einstein condensate occurs at a critical temperature $T_C$ or critical density $n_C$.
\[N \geq \int_0^\infty{\frac{d\epsilon\ g(\epsilon)}{e^{\epsilon/\tau} - 1}} = 2.612\frac{V}{\ell_Q^3}\]\[T_C = \frac{2\pi\hbar}{k_B m}\left(\frac{N}{2.612 V}\right)^{2/3}\]\[n_C = \frac{2.612}{\ell_Q^3}\]

Result 3.7.5  For $T\leq T_C$, the pressure can be approximated is terms of this expansion.
\[P \approx f_{5/2}^{+1}(1) \frac{k_B T}{\ell_Q^3}\]

Definition 3.7.6  The Riemann zeta function denoted $\zeta(s)$ is a function $\zeta:\mathbb{C}\to\mathbb{C}$ defined by
\[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{\Gamma(s)}\int_0^\infty{\frac{x^{s-1}}{e^x-1}dx}\]
where $\Gamma(s) = \int_0^\infty{x^{s-1}e^{-x}dx}$ is the gamma function.

3.8 Photon GasR61E

Definition 3.8.1  A photon gas is a system consisting of photons in a box with the following energy levels and two polarization modes $\lambda = \pm 1$. This system can be considered at as a Bose gas with $g_s=2$ and $\epsilon(p) = pc$ and $\mu = 0$.
\[k = \frac{2\pi}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]\[\epsilon_k = \hbar c k = \frac{2\pi\hbar c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]

Result 3.8.2  For a photon gas in canonical ensemble, the partition function, Helmholtz free energy, pressure, energy and entropy are given by the following equations.
\[z_{k,\lambda} = \sum_{n_{k,\lambda}=0}^\infty{e^{-\beta\epsilon_k}} = \frac{1}{1-e^{-\beta\hbar\omega_k}}\]\[z = \prod_{k,\lambda}z_{k,\lambda} =\prod_{k,\lambda}\frac{1}{1-e^{-\beta\hbar ck}}\]\[F = \frac{\pi^2 Vk_B^4 T^4}{45(\hbar c)^3},\quad P = \frac{\pi^2 k_B^4 T^4}{45(\hbar c)^3}\]\[E = \frac{3\pi^2 Vk_B^4T^4}{45(\hbar c)^3} = 3 PV,\quad S = \frac{4\pi^2Vk_B^4 T^3}{45(\hbar c)^3}\]

Law 3.8.3  Plank's law states that the spectral energy density $\mu(\omega)$ can be written in terms of temperature and frequency $\omega$.
\[\frac{E}{V} = \int d\omega \mu(\omega) = \frac{\hbar V}{\pi^2 c^3}\int{\frac{d\omega\ \omega^3}{e^{\beta\hbar\omega} - 1}}\]\[\mu(\omega) = \frac{\hbar}{\pi^2 c^3}\frac{\omega^3}{e^{\beta\hbar\omega} - 1}\]

Definition 3.8.4  A black body is a material that perfectly absorbs electromagnetic radiation of all frequencies.

Definition 3.8.5  The absorptivity $\alpha$ of a material is the fraction of photons absorbed by the object at temperature $T$.

Definition 3.8.6  The emmisivity $e$ of a material is the fraction of black body radiation emitted by an object at temperature $T$.

Law 3.8.7  Kirchoff's law states that the absorptivity $a$ and emmisivity $e$ of a material are equal $e=a$.

3.9 Phonon GasKWHN

Definition 3.9.1  Phonons are a system of $3N$ harmonic oscillators with frequencies $\omega_i$.
\[E = \sum_{i=1}^{3N}{\epsilon_i} = \sum_{i=1}^{3N}{n_i\hbar\omega_i}\]

Definition 3.9.2  The Einstein model of a phonon gas simplifies the harmonic oscillators to all have the same frequency $\omega$.

Definition 3.9.3  The Debye frequency denoted $\omega_D$ is the maximum frequency in a material due to the lattice spacing.

Definition 3.9.4  The Debye model models phonons as an elastic wave with speed of sound $c_s$ and a maximum frequency $\omega_D$.
\[\omega_{n_x,n_y,n_z} = c_sk = \frac{2\pi c_s}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}\]

Result 3.9.5  The Debye frequency for phonons $\omega_D$ in a solid with speed of sound $c_s$ in $m$-dimensional space can be derived from the number of particles $N$.
\[mN = m\sum_{n}\Theta(\omega_D-\omega_n) = \frac{mL^m}{(2\pi c_s)^m}\int d\Omega_m\int_0^{\omega_D}\omega^{m-1}d\omega\]\[\omega_D^{m} = 2m\pi^{m-1}\frac{N}{L^m}c_s^m\]

Result 3.9.6  The total energy $E$ for phonons in a solid with speed of sound $c_s$ in $m$-dimensional space can be derived in terms of the Debye frequency $\omega_D$.
\[E = m\sum_{n}\epsilon_n \to \frac{mL^m}{(2\pi c_s)^m}\int d\Omega_m\int_0^{\omega_D}\frac{\hbar\omega^m}{e^{\hbar\omega/\tau}-1}d\omega\]

4 Interacting Systems and Phase TransitionsJKED

4.1 Virial Expansion7AT9

Definition 4.1.1  The virial expansion is a perturbative approach to finding an approximate canonical partition function for a system with Hamiltonian $H$ that consists of a Hamiltonian $H_0$ with a known partition function $z_0$ and a small perturbation Hamiltonian $V$ the following holds.
\[H=H_0+V\]\[z = z_0\left(1 + \sum_{n=1}^\infty\frac{(-\beta)^n}{n!}\langle V^n\rangle_0\right)\]\[\langle V^n \rangle_0 = \frac{\text{Tr}(e^{-\beta H_0}V^n)}{\text{Tr}(e^{-\beta H_0})}\]

Definition 4.1.2  An interacting ideal gas is a perturbative system for an ideal gas $(n\ell_Q^3 << 1)$ with some small interaction potential $u(r)$ between particles.
\[H = H_0 + V = \sum_{i=1}^N{\frac{p_i^2}{2m}} + \sum_{i<j}u(|r_i-r_j|)\]

Result 4.1.3  The virial expansion for an interacting ideal gas approximates the equation of state for an interacting ideal gas in the grand canonical ensemble.
\[\frac{PV}{N\tau} = 1 + \sum_{m=1}^{\infty}{a_m(\tau)\left(n\ell_Q^3\right)^m}\]\[\mathscr{z} = \sum_{N=0}^\infty{\frac{\mathfrak{z}^N}{N!\ell_Q^{3N}}Q_N}\]\[Q_N = \int d^{3N}r e^{-\beta \sum_{i<j}u(|r_i-r_j|)} \]\[Q_1 = \int d^3 N = V, \quad Q_2 = \int d^3 r_1 d^3 r_2 e^{-\beta u(|r_i-r_j|)} = \int d^3R \int d\Omega_3 \int_0^\infty r^2 e^{-\beta u(r)} dr\]\[a_1 = \frac{Q_1}{V} = 1,\quad a_2 = \frac{-1}{2\ell_Q^3 V}(Q_2-Q_1^2) = \frac{-2\pi}{\ell_Q^3}\int_0^\infty r^2 (e^{-\beta u(r)}-1) dr\]\[\frac{PV}{N\tau} \approx 1 - \frac{2\pi N}{V}\int_0^\infty r^2 (e^{-\beta u(r)}-1) dr\]

4.2 Van Der Waals Gas737E

Definition 4.2.1  A Van der Waals gas is an interacting gas system with interacting potential $u(r)$ defined by
\[u(r) = \begin{cases}\infty & r<r_0\\ -u_0\left(\frac{r_0}{r}\right)^6 & r\geq r_0\end{cases}\]

Result 4.2.2  The virial expansion for a Van der Waals gas approximates the equation of states in the grand canonical ensemble.
\[a_2 = \frac{2\pi r_0^3}{3 \ell_Q^3} \left(1-\frac{u_0}{\tau}\right)\]\[\frac{PV}{N\tau} \approx a_1 + a_2 \frac{N\ell_Q^3}{V} = 1 + \frac{2\pi r_0^3N}{3 V} \left(1-\frac{u_0}{\tau}\right)\]\[b = \frac{2\pi r_0^3N}{3V},\quad a=bu_0 = \frac{2\pi r_0^3 u_0}{3}\]\[\frac{PV}{N\tau} \approx 1 + \frac{bN}{V}\left(1-\frac{a}{b\tau}\right)\]
For a dilute gas where $\frac{V}{N}>> b$, we have the van der Waal equation of state
\[\left(P + \frac{aN^2}{V^2}\right)\left(\frac{V}{N} - b\right) = \tau\]

4.3 Phase TransistionsR8N0

Definition 4.3.1  A phase transition is any non analytic behavior of thermodynamic quantities, such as discontinuities or divergences.

Definition 4.3.2  An order parameter is a thermodynamic quantity that distinguishes the phases of a phase transition.

Definition 4.3.3  A discontinuous phase transition or a first order phase transition is a phase transition characterized by a discontinuity of the order parameter.

Definition 4.3.4  A continuous phase transition or a second order phase transition is a discontinuity in at least one of the derivative of the order parameter.

Definition 4.3.5  A critical exponent is the exponent $\beta$ that describes the limiting behavior $|t|^\beta$ of an order parameter $x(t)$ in terms of $t = \frac{T-T_C}{T_C}$.
\[x(t)\sim |t|^\beta\]

Theorem 4.3.6  The fluctuation dissipation theorem states that the magnetic susceptibility order parameter $\xi_T$ for the ferromagnet to paramagnet transition is related to spacial fluctuation in the magnetization $m(r)$.
\[\xi_T = \beta V\int d^3r \langle m(r)m(0)\rangle\]

4.4 Landau Ginzburg TheoryH55M

Definition 4.4.1  Landau Ginzburg theory rewrites the partition function in terms of a larger scale effective hamiltonian $H_{eff}$ that depends on a slowly varying function $m(r)$.
\[\mathscr{z} = \int \prod_{I = 1}^{N_{cells}} dm(r_I) e^{-\beta H_{eff}[m(r)]}\]\[e^{-\beta H_{eff}[m(r)]} = \sum_{\{S_i\}} e^{-\beta H} \prod_I \delta\left(m(r_I)-\frac{1}{V_I}\sum_{i\in I}S_I\right)\]
however computation of exact $H_{eff}$ is not practically possible for large systems.

Result 4.4.2  The Landau Ginzburg rules describe how to approximate with mean field theory the form of $H_{eff}$.

1. Locality states that we should be able to write the effective Hamiltonian $H_{eff}$ in terms of a local energy density $\Phi$.
   \[\beta H_{eff} = \int d^dr \Phi[m(r), \nabla m(r), \dots]\]
2. Analytic/Polynomial Expansion in m states that the $\Phi$ can be expanded as a polynomial in terms of $m(r)$,
   \[\Phi = \text{const} + q_1m + q_2 m^2 +\dots + \kappa(\nabla m)^2 + \dots\]
3. $H_{eff}$ should respect underlying symmetries. For symmetric systems $H_{eff}[m] = H_{eff}[-m]$ so
   \[q_1 = q_3 = \dots = 0\]
4. Stability the coefficient of highest power in $m$ must be positive to avoid the infinite limit.
5. Coefficient of gradient term states that the gradient term $\kappa\nabla m$ should be positive because it is energetically favorable for nearby states to be aligned.

Result 4.4.3  Landau Ginzburg Theory for Magnetization states that
\[\beta H_{eff}\approx \beta F_0 + \int d^dr \left[ q_2 m^2 + q_4 m^4 + \frac{\kappa}{2}\left(\nabla m\right)^2 \right], \quad q_4 > 0, \kappa > 0\]\[\beta H_{eff} \approx \beta F_0 V (q_2 m^2 + q_4m^4 - Hm)\]\[q_2 > 0 \text{ for } T>T_C, \quad q_2 < 0 \text{ for } T < T_c \text{ and }\bar{m}\neq 0\]\[q_2(t) \approx \frac{a}{2}t,\quad q_4 \approx b > 0\]\[m(t) \sim |t|^\beta,\quad \beta = \frac{1}{2}\]

Quantum Mechanics 1

Quantum Mechanics 2

ElectromagnetismEDRP

Electromagnetism in Free SpaceAHC1

1 IntroductionE4KM

1.1 NotationPD4J

Definition 1.1.1  The Cartesian coordinates system is a coordinates system that uses three coordinates $(x,y,z)$ to define a point in three dimensional space. A position vector $\vec{r}$ can be written in terms of the elementary basis vectors $\hat{x},\hat{y},\hat{z}$,
\[\vec{r}=(x,y,z)=x \hat{x} + y \hat{y} + z \hat{z}\]

Definition 1.1.2  The spherical coordinate system is a coordinate system that uses three coordinates $(r,\theta,\phi)$ to define a point in three dimensional space. For a particular position we also define the basis vectors $\hat{r},\hat{\theta},\hat{\phi}$,

Definition 1.1.3  The cylindrical coordinate system is a coordinate system that uses the three coordinates $(s,\phi,z)$ to define a point in three dimensional space. For a particular position we also define the basis vectors $\hat{s},\hat{\phi},\hat{z}$,

Definition 1.1.4  The Kronecker delta $\delta_{ij} = \left\{\begin{array}{lr}
1, & \text{if } i = j\\
0, & \text{if } i\neq j
\end{array}\right\}$.

Definition 1.1.5  The Levi-Civita symbol $\varepsilon_{a_1,a_2,\dots,a_n} = \left\{\begin{array}{rl}
+1, & \text{if } (a_1,a_2,\dots,a_n)\text{ is an even permutation of }(1,2,\dots,n)\\
-1, & \text{if } (a_1,a_2,\dots,a_n)\text{ is an odd permutation of }(1,2,\dots,n)\\
0, & \text{otherwise}
\end{array}\right\}$

Definition 1.1.6  The dot product denoted $\vec{a}\cdot\vec{b}$ of two vectors $\vec{a}$ and $\vec{b}$ is the sum of the products of there components in any orthonormal basis.
\[\vec{a}\cdot\vec{b} = \sum_{i}a_ib_i\]

Definition 1.1.7  The cross product of two vectors $\vec{a}$ and $\vec{b}$ is defined by the following sums of the products of there components in any orthonormal basis $\vec{e_1},\vec{e_2},\dots,\vec{e_n}$.
\[\vec{a}\times\vec{b} = \sum_i\sum_j\sum_k\varepsilon_{ijk}a_jb_k \vec{e_i}\]

Definition 1.1.8  The gradient denoted $\nabla F$ of a differentiable scalar field $F$ is the vector field defined by the partial derivatives of the scalar field.
\[\nabla F = \frac{\partial F}{\partial x}\hat{x} + \frac{\partial F}{\partial y}\hat{y} + \frac{\partial F}{\partial z}\hat{z}=\left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right)\]\[\nabla F = \frac{\partial F}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial F}{\partial \theta}\hat{\theta} + \frac{1}{r\sin\theta} \frac{\partial F}{\partial \phi}\hat{\phi}\]

Definition 1.1.9  The divergence denoted $\nabla \cdot \vec{F}$ of a differentiable vector field $\vec{F}$ is a scalar field defined by the sum of partial derivatives of the components of the vector field.
\[\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\]\[\nabla \cdot \vec{F} = \frac{1}{r^2}\frac{\partial (r^2F_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta} \left(F_\theta \sin\theta\right) + \frac{1}{\sin\theta}\frac{\partial F_\phi}{\partial \phi}\]\[\nabla \cdot \vec{F} = \frac{1}{s}\frac{\partial (sF_s)}{\partial s} + \frac{1}{s} \frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z}\]

Definition 1.1.10  The curl denoted $\nabla \cross \vec{F}$ of a differentiable vector field $\vec{F}$ is a vector field defined in terms of the following partial derivatives.
\[\nabla \times \vec{F}=\det\begin{pmatrix}
\hat{x} & \hat{y} & \hat{z}\\
\frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\
F_x & F_y & F_z
\end{pmatrix}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\hat{y}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\hat{z}\]\[\nabla \times \vec{F} = \frac{1}{r\sin\theta}\left( \frac{\partial }{\partial\theta}(F_\phi \sin\theta) - \frac{\partial F_\theta}{\partial \phi} \right)\hat{r} + \frac{1}{r}\left( \frac{1}{\sin\theta}\frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}(rF_\phi)\right)\hat{\theta} + \frac{1}{r}\left(\frac{\partial }{\partial r}(rF_\theta)-\frac{\partial F_r}{\partial \theta}\right)\]\[\nabla \times \vec{F} = \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right)\hat{s} + \left( \frac{\partial F_s}{\partial z} - \frac{\partial F_z}{\partial s} \right)\hat{\phi} + \frac{1}{s}\left(\frac{\partial (sF_\phi)}{\partial s} - \frac{\partial F_s}{\partial \phi} \right)\hat{z}\]

Definition 1.1.11  The Laplace denoted $\nabla^2 F$ of a differentiable scalar field $F$ is the divergence of the gradient of $F$.
\[\nabla^2F = \nabla\cdot\nabla F = \frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2}\]

Proposition 1.1.12  The curl of a gradient is zero, that is $\nabla\times(\nabla F)=0$ for any scalar field $F$.

Proposition 1.1.13  The divergence of a curl is zero, that is $\nabla\cdot(\nabla \times \vec{F})=0$ for any vector field $\vec{F}$.

Proposition 1.1.14  The curl of curl can be written in terms of gradients and divergences for any vector field $\vec{F}$.
\[\nabla\times\left(\nabla\times\vec{F}\right) = \nabla\left( \nabla\cdot\vec{F} \right) - \nabla\cdot\nabla\vec{F} = \nabla\left( \nabla\cdot\vec{F} \right) - \nabla^2\vec{F}\]

Theorem 1.1.15  The gradient theorem states that gradients are path independent, that is for any differentiable scalar field $F$,
\[\int_\vec{a}^\vec{b}(\nabla F)\cdot d\vec{\ell} = F(\vec{b})-F(\vec{a})\]

Theorem 1.1.16  The divergence theorem or Gauss's theorem states that the surface integral of continuously differentiable vector field $\vec{F}$ dotted with the normal vector $d\vec{S}$ over the piecewise smooth boundary $S$ of a volume $V$ is the volume integral of the divergence of $\vec{F}$ over the volume $V$.
\[\iint_S \vec{F}\cdot d\vec{S} = \iiint_V(\nabla\cdot\vec{F})dV\]

Theorem 1.1.17  The curl theorem or Stokes' theorem states that the loop integral of a continuously differentiable vector field $\vec{F}$ along the boundary path $P$ of a smooth oriented surface $S$ is the surface integral of the curl of $\vec{F}$ over the surface $S$.
\[\int_P\vec{F}\cdot d\vec{P} = \iint_S(\nabla\times\vec{F})\cdot d\vec{S}\]

Definition 1.1.18  The Dirac delta denoted $\delta(x)$ is the notation for a distribution peaked infinitesimally around zero.
\[\delta(x) = \lim_{a\to 0} \frac{1}{|a|\sqrt{\pi}}e^{-(x/a)^2}\]

1.2 SI Units8ZFC

Definition 1.2.1  The SI unit system is the most popular system of units that uses the fundamental units of seconds, meters, kilograms, ampere and Kelvin to derive a system of units to describe the universe.

Definition 1.2.2  A second (s) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

Definition 1.2.3  A meter (m) is the SI unit of distance that is exactly the distance light travels in $1/299792458$ seconds.

Definition 1.2.4  A kilogram (kg) is the SI unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−34}\text{kg } \text{m}^2\text{s}^{−1}$.

Definition 1.2.5  An ampere (A) is the SI unit of current that is exactly the flow of $10^{19}/1.602176634$ protons per second.

Definition 1.2.6  A Kelvin (K) is the SI unit of absolute temperature defined exactly by fixing Boltzmann's constant $k=1.380649\times 10^{-23}\text{kg }\text{m}^2 \text{s}^{-2}\text{K}^{-1}$.

Definition 1.2.7  A Coulomb (C) is the SI unit of charge defined by $\text{C} = \text{A s}$ or exactly $10^{19}/1.602176634$ protons.

Definition 1.2.8  A Newton (N) is the SI unit of force defined by $\text{N} = \text{kg m}/\text{s}^{2}$.

Definition 1.2.9  A Joule (J) is the SI unit of energy defined by $\text{J} = \text{N m} = \text{kg }\text{m}^{2}/\text{s}^{2}$.

Definition 1.2.10  A Watt (w) is the SI unit of power defined by $\text{w} = \text{J}/\text{s}$.

Definition 1.2.11  A Pascal (Pa) is the SI unit of pressure defined by $\text{Pa}=\text{N}/\text{m}^2 = \text{J}/\text{m}^3 = \text{kg }\text{m}^{-1}\text{s}^{-2}$.

Definition 1.2.12  A Volt (V) is the SI unit of electric potential defined by $\text{V} = \text{J}/\text{C} = \text{w}/\text{A} = \text{kg }\text{m}^2\text{s}^{-3}\text{A}^{-1}$.

Definition 1.2.13  A Volt per Meter (V/m) is the SI unit of electric field defined by $\text{V}/\text{m} = \text{N}/\text{C} = \text{kg m}\text{s}^{-2}\text{A}^{-1}$.

Definition 1.2.14  A Telsa (T) is the SI unit of magnetic field defined by $\text{T} = \text{V s}/\text{m}^2 = \text{kg }\text{s}^{-2}\text{A}^{-1}$.

Definition 1.2.15  The fine structure constant denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value¹:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

Law 1.2.16  Maxwell's Equations are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]\[\nabla\cdot \vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

Definition 1.2.17  The vacuum permittivity $\varepsilon_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\varepsilon_0 = \frac{e^2}{2\alpha hc}\]

Definition 1.2.18  The vacuum permeability $\mu_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\mu_0 = \frac{2\alpha h}{e^2c}\]

Result 1.2.19  The product of vacuum permittivity and vacuum permeability is the reciprocal of the speed of light squared.
\[\varepsilon_0\mu_0 = \frac{1}{c^2}\]

1.3 Maxwell's EquationsCRAD

Law 1.3.1  Maxwell's Equations are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]\[\nabla\cdot \vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

Definition 1.3.2  The vacuum permittivity $\varepsilon_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\varepsilon_0 = \frac{e^2}{2\alpha hc}\]

Definition 1.3.3  The vacuum permeability $\mu_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\mu_0 = \frac{2\alpha h}{e^2c}\]

Result 1.3.4  The product of vacuum permittivity and vacuum permeability is the reciprocal of the speed of light squared.
\[\varepsilon_0\mu_0 = \frac{1}{c^2}\]

2 ElectrostaticsJ0C0

2.1 Electric Field70R3

Definition 2.1.1  The electric field denoted $\vec{E}(\vec{r})$ is a vector field of the force that would be felt by a test charge at a point in space. The units of electric field are Newtons per Coulomb denoted $N/C$. For a charge $q$ at position $\vec{r}$ the force $\vec{F}$ from electric field $\vec{E}$ can be calculated with the following equation:
\[\vec{F} = q\vec{E}(\vec{r})\]

Definition 2.1.2  The electric displacement field denoted $\vec{D}$ is defined in terms of the electric field $\vec{E}$ the polarization $\vec{P}$.
\[\vec{D} = \varepsilon_0\vec{E} + \vec{P}\]

Definition 2.1.3  The polarization denoted $\vec{P}$ is the electric dipole moment per unit volume of the bound charge density $\rho_b$ in a material.
\[\vec{P} = \frac{d\vec{p}}{dV}\]\[- \nabla\cdot\vec{P} = \rho_b\]

Law 2.1.4  Coulomb's law states that the force $\vec{F}$ on a point charge $q_1$ located at $\vec{r}_1$ due to another point charge $q_2$ located at $\vec{r}_2$ can be obtained with the following equation: $\newcommand\abs[1]{\left|#1\right|}$
\[\vec{F} = \frac{1}{4\pi\varepsilon_0}q_1q_2\frac{\vec{r}_1-\vec{r}_2}{\abs{\vec{r}_1-\vec{r}_2}^3}\]

Result 2.1.5  The electric field of a point charge $q_1$ located at $\vec{r}_1$ can be directly obtained from coulomb's law.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0}q_1\frac{\vec{r}-\vec{r}_1}{\abs{\vec{r}-\vec{r}_1}^3}\]

Result 2.1.6  The electric field of many point charges $q_1,\dots,q_n$ located at $\vec{r}_1,\dots,\vec{v}_n$ can be directly obtained from coulomb's law.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n{q_i}\frac{\vec{r}-\vec{r}_i}{\abs{\vec{r}-\vec{r}_i}^3}\]

Definition 2.1.7  A charge density denoted $\rho(\vec{r})$ is the function whose integral represents the charge in a region of space.

Result 2.1.8  The electric field of a charge density $\rho$ can be obtained by direct integration.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int{ \rho(\vec{r}') \frac{\vec{r}-\vec{r}'}{\abs{\vec{r}-\vec{r}'}^3} d^3r'}\]

Law 2.1.9  Gauss's law states that states that the surface integral of the electric field $\vec{E}$ dotted with the normal vector $\vec{n}$ over a piecewise smooth boundary $S$ of a volume $V$ is the volume integral of the charge density $\rho$ over a volume $V$.
\[\iint_S (\vec{E}\cdot \vec{n}) d^2r = \frac{1}{\varepsilon_0}\iiint_V\rho(\vec{r})d^3r\]

Result 2.1.10  Gauss's law for discrete charges states that for discrete charges the volume integral can be simplified into a sum of the chargest inside the volume.
\[\iint_S (\vec{E}\cdot \vec{n}) d^2r = \frac{1}{\varepsilon_0}\sum_i{q_i}\]

2.2 Electric Potential35AD

Definition 2.2.1  The electric potential or voltage denoted $V(\vec{r},t)$ is a scalar potential such that the negative gradient is the electric field in electrostatic systems.
\[\vec{E} = -\nabla V - \frac{\partial \vec{A}}{\partial t}\]

Result 2.2.2  The electric potential of a charge density can be obtained by direct integration.$\newcommand\abs[1]{\left|#1\right|}$
\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int{ \frac{\rho(\vec{r}')}{\abs{\vec{r}-\vec{r}'}} d^3r'}\]

Result 2.2.3  The electric fields near a surface charge distribution are discontinuous in the direction normal $\vec{n}$ to the surface charge density $\sigma$.
\[(\vec{E}_{out}-\vec{E}_{in})\cdot\vec{n} = \sigma(\vec{r})/\varepsilon_0\]

Result 2.2.4  The electric potentials near a surface dipole layer, with dipole moment $\vec{D}(\vec{r})$ in the direction normal to the surface, are discontinuous proportional to the dipole moment $D(\vec{r})$.
\[V_2-V_1 = D(\vec{r})/\varepsilon_0\]

Result 2.2.5  The potential energy of many point charges $W$ is the total energy of a set of many point charges $q_1,\dots,q_n$ at positions $\vec{r}_1,\dots,\vec{r}_n$ due to the forces acting between them.$\newcommand\abs[1]{\left|#1\right|}$
\[W = \frac{1}{8\pi\varepsilon_0}\sum_i\sum_j\frac{q_iq_j}{\abs{\vec{r}_i-\vec{r}_j}}\]

Result 2.2.6  The potential energy of a charge distribution $W$ is the total energy of a charge distribution $\rho(\vec{r})$.$\newcommand\abs[1]{\left|#1\right|}$
\[W = \frac{1}{8\pi\varepsilon_0}\int\int\frac{\rho(\vec{r})\rho(\vec{r}')}{\abs{\vec{r}-\vec{r}'}}d^3r\ d^3r' = \frac{\varepsilon_0}{2}\int\abs{\vec{E}}^2d^3r\]

2.3 Spherical HarmonicsW5EF

Definition 2.3.1  The Legendre polynomials $P_\ell(x)$ are a set of polynomials defined on the interval $-1\leq x \leq 1$ for $\ell \in \{0,1,2,\dots\}$ by the following expression.
\[P_\ell(x) = \frac{1}{2^\ell}\]

Definition 2.3.2  The associated Legendre polynomials $P_\ell^m(x)$ are a set of polynomials defined on the interval $-1\leq x \leq 1$ for $\ell \in \{0,1,2,3,\dots\}$ and $m \in \{-\ell,-\ell+1,\dots,0,\dots,\ell-1,\ell\}$ by the following expression.
\[P_\ell^m(x) = \frac{(-1)^{m}}{2^\ell\ell!}(1-x^2)^{m/2}\frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell\]

Definition 2.3.3  The spherical harmonics $Y_{\ell,m}(\theta,\phi)$ are a set of spherical functions defined for $\ell \in \{0,1,2,3,\dots\}$ and $m \in \{-\ell,-\ell+1,\dots,0,\dots,\ell-1,\ell\}$ that forms an orthonormal bases for the set of complex spherical functions.
\[Y_{\ell,m}(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos\theta)e^{im\phi}\]

Result 2.3.4  Properties of Spherical Harmonics: The spherical harmonics $Y_{\ell,m}$ and $Y_{\ell, -m}$ are related by the following relation,
\[Y_{\ell,-m}(\theta,\phi) = (-1)^{m}Y_{\ell,m}^*(\theta, \phi)\]
Spherical harmonics from an orthonormal basis with the following orthogonality and normalization conditions,
\[\int_0^{2\pi}\int_0^\pi Y^*_{\ell,m'}(\theta,\phi)Y_{\ell,m}(\theta,\phi) \sin\theta\ d\theta\ d\phi = \delta_{\ell',\ell}\delta_{m',m}\]
Spherical harmonics span the set of all complex spherical functions as show by the following completeness relation,
\[\sum_{\ell = 0}^\infty \sum_{m=-\ell}^\ell Y_{\ell,m}^*(\theta',\phi')Y_{\ell,m}(\theta,\phi) = \delta(\phi-\phi')\delta(\cos\theta-\cos\theta')\]

Table 2.3.5  Table of Spherical Harmoincs
\[Y_{0,0} = \frac{1}{\sqrt{4\pi}}\]\[Y_{1,1} = -\sqrt{\frac{3}{8\pi}}\sin\theta\ e^{i\phi}\]\[Y_{1,0} = \sqrt{\frac{3}{4\pi}}\cos\theta\]\[Y_{2,2} = \frac{1}{4}\sqrt{\frac{15}{2\pi}} \sin^2\theta\ e^{2i\phi}\]\[Y_{2,1} = -\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta\ e^{i\phi}\]\[Y_{2,0} = \sqrt{\frac{5}{4\pi}}\left(\frac{3}{2}\cos^2\theta - \frac{1}{2}\right)\]

2.4 Electric Multipole Expansion6AJ3

Theorem 2.4.1  The multipole expansion theorem states that for a charge distribution localized within a sphere, the electric potential outside the sphere can be written in terms of spherical harmonics.
\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell\frac{4\pi}{2\ell+1}q_{\ell,m}\frac{Y_{\ell,m}(\theta,\phi)}{r^{\ell+1}}\]\[q_{\ell,m} = \int Y^*_{\ell,m}(\theta,\phi)r^{\ell}\rho(\vec{r})d^3r\]

Result 2.4.2  The multipole moment $q_{\ell,m}$ and $q_{\ell, -m}$ are related by the relation $q_{\ell,-m} = (-1)^{m}q^*_{\ell,m}$.

Definition 2.4.3  The monopole moment $q$ of a charge distribution $\rho(\vec{r})$ is simply the total charge.
\[q = \int \rho(\vec{r}) d^3r\]

Definition 2.4.4  The dipole moment $\vec{p}$ of a charge distribution $\rho(\vec{r})$ is defined with the following integral.
\[\vec{p} = \int\vec{r}\rho(\vec{r})d^3r\]

Definition 2.4.5  The quadrupole moment $Q$ of a charge distribution $\rho(\vec{r})$ is a $3$ by $3$ matrix with components $Q_{j,k}$ defined with the following integral.
\[Q_{j,k} = \int \left[3r_jr_k-r^2\delta_{j,k}\right]\rho(\vec{r})d^3r\]

Result 2.4.6  The multipole expansion of a charge distribution can be written in terms of the multiple moments of the charge distribution. The first 6 terms of the multiple expansion are written below in terms of the monopole moment $q$, the dipole moment $\vec{p}$ and the quadrupole moment $Q$ of a charge distribution $\rho(\vec{r})$.
\[q_{0,0} = \frac{1}{\sqrt{4\pi}}\int\rho(\vec{r})d^3r = \frac{1}{\sqrt{4\pi}}q\]\[q_{1,1} = -\sqrt{\frac{3}{8\pi}}\int(x-iy)\rho(\vec{r})d^3r = -\sqrt{\frac{3}{8\pi}}(p_x-ip_y)\]\[q_{1,0} = \sqrt{\frac{3}{4\pi}}\int z\rho(\vec{r})d^3r = \sqrt{\frac{3}{4\pi}}p_z\]\[q_{2,2} = \frac{1}{4}\sqrt{\frac{15}{2\pi}}\int(x-iy)\rho(\vec{r})d^3r = \frac{1}{12}\sqrt{\frac{15}{2\pi}}(Q_{1,1}-2iQ_{1,2}-Q_{2,2})\]\[q_{2,1} = -\sqrt{\frac{15}{8\pi}}\int z(x-iy)\rho(\vec{r})d^3r = -\frac{1}{3}\sqrt{\frac{15}{8\pi}}(Q_{1,3}-iQ_{2,3})\]\[q_{2,0} = \frac{1}{2}\sqrt{\frac{5}{4\pi}}\int(3z^2-r^2)\rho(\vec{r})d^3r = \frac{1}{2}\sqrt{\frac{5}{4\pi}}Q_{3,3}\]

2.5 Cartesian Separation of VariablesD60D

Theorem 2.5.1  The solution $y(x)$ to a second order linear differential equation of the form
\[a\frac{d^2 y(x)}{dx^2} + b\frac{d y(x)}{dx} + c y(x) = 0\]
is $y=Ae^{r_1x} + Be^{r_2x}$ for some constants $A,B\in\mathbb{C}$ and where $r_1,r_2\in\mathbb{C}$ are the two solutions to the quadratic equation $ar^2 + br + c = 0$.

Corollary 2.5.2  An equation of the form $\frac{1}{y(x)}\frac{d^2 y(x)}{dx^2} = k$ has solution $y(x) = Ae^{\sqrt{k}x} + Be^{-\sqrt{k}x}$ for some constants $A,B\in\mathbb{C}$.

Definition 2.5.3  Separation of variables is a technique solve differential equations by algebraically separating the equation into independent one dimensional differential equations.

Definition 2.5.4  The Laplace equation is for a scalar function $V(\vec{r})$ is the second-order partial differential equation defined by
\[\nabla^2V(\vec{r}) = 0\]
where $\nabla^2$ is the Laplace operator. This equation describes how an electrostatic potential function behaves in a region with zero charge density.

Theorem 2.5.5  The uniqueness theorem for the Laplace equation states that for a

Result 2.5.6  The Laplace equation written in Cartesian coordinates for a potential $V(x,y,z)$ is
\[\nabla^2 V(\vec{r}) = \frac{\partial^2 V(\vec{r})}{\partial x^2} + \frac{\partial^2 V(\vec{r})}{\partial y^2} + \frac{\partial^2 V(\vec{r})}{\partial z^2} = 0\]

Result 2.5.7  Applying separation of variables for the Laplace equation in Cartesian coordinates and assuming that the solution is in product form $V(x,y,z) = X(x)Y(y)Z(z)$, produces the following differential equations.
\[\frac{1}{X}\frac{d^2 X}{d x^2} + \frac{1}{Y}\frac{d^2 Y}{d y^2} + \frac{1}{Z}\frac{d^2 Z}{d z^2} = 0\]\[\frac{1}{X}\frac{d^2 X}{d x^2} = k_x,\quad \frac{1}{Y}\frac{d^2 Y}{d y^2} = k_y,\quad \frac{1}{Z}\frac{d^2 Z}{d z^2} = k_z\]

2.6 Spherical Separation of VariablesRHZ7

Result 2.6.1  The Laplace equation in spherical coordinates for a potential $V(r,\theta,\phi)$ is
\[\frac{1}{r}\frac{\partial^2}{\partial r^2}(rV(\vec{r})) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\ \frac{\partial V(\vec{r})}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta} \frac{\partial^2V(\vec{r})}{\partial \phi} = 0\]

Result 2.6.2  Applying separation of variables for the Laplace equation in Spherical coordinates and assuming that the solution is in product form $V(r,\theta,\phi) = \frac{U(r)}{r}P(\theta)Q(\phi)$, produces the following differential equation.
\[r^2\sin^2\theta\left[ \frac{1}{U}\frac{d^2 U}{dr^2} + \frac{1}{Pr^2\sin\theta}\frac{d}{d\theta}\left(\sin\theta\ \frac{dP}{d\theta}\right) \right] + \frac{1}{Q}\frac{d^2Q}{d\phi^2} = 0\]
From this it can be shown that any solution can be written in terms of spherical harmonics with coefficients $A_{\ell,m},B_{\ell,m}\in\mathbb{R}$.
\[V(r,\theta,\phi) = \sum_{\ell=0}^\infty{\sum_{m=-\ell}^\ell{\left[ A_{\ell,m} r^\ell + \frac{B_{\ell,m}}{r^{\ell+1}} \right]Y_{\ell,m}(\theta,\phi)}}\]\[A_{\ell,m}a^{\ell} + \frac{B_{\ell,m}}{a^{\ell+1}} = \int Y^*_{\ell,m}(\theta,\phi)V(r=a,\theta,\phi)d\Omega\]

Corollary 2.6.3  A Legendre polynomial solution to the Laplace equation in spherical coordinates for systems with azimuthal symmetry can be written in terms of Legendre polynomials $P_\ell$ with coefficients $A_{\ell,m},B_{\ell,m}\in\mathbb{R}$.
\[V(r,\theta) = \sum_{\ell = 0}^\infty\left[A_{\ell,m} r^\ell + \frac{B_{\ell,m}}{r^{\ell+1}}\right]P_\ell(\cos\theta)\]

2.7 Cylindrical Separation of VariablesJW99

Result 2.7.1  The Laplace equation in cylindrical coordinates for a potential $V(s,\phi,z)$ is
\[\frac{\partial^2 V(\vec{r})}{\partial s^2} + \frac{1}{s}\frac{\partial V(\vec{r})}{\partial s} + \frac{1}{s^2}\frac{\partial^2 V(\vec{r})}{d\phi^2} + \frac{\partial^2 V(\vec{r})}{\partial z^2} = 0\]

Result 2.7.2  Applying separation of variables for the Laplace equation in cylindrical coordinates and assuming that the solution is in product form $V(s,\phi,z) = S(s)Q(\phi)Z(z)$, produces the following system of differential equations for some constants $k,v\in\mathbb{R}$.
\[\frac{1}{Z(z)}\frac{\partial^2 Z(z)}{\partial z^2} = k^2\]\[\frac{1}{Q(\phi)}\frac{\partial^2 Q(\phi)}{\partial \phi^2} = -v^2\]\[\frac{\partial^2 S(s)}{\partial s^2} + \frac{1}{s}\frac{\partial S(s)}{\partial s} + \left(k^2 - \frac{v^2}{s^2}\right)S(s) = 0\]

Definition 2.7.3  The Bessel functions are the canonical solutions $y(x)$ of Bessel's differential equation, defined below for some complex number $\alpha$.
\[x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y = 0\]

Corollary 2.7.4  The third differential equation in separation of variables for the Laplace equation in cylindrical coordinates can be written as Bessel's differential equation where $x=ks$ and $\alpha = v$,
\[x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - v^2)y = 0\]\[\frac{d^2y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + \left(1 - \frac{v^2}{x^2}\right)y = 0\]

Definition 2.7.5  The Bessel functions of the first kind denoted $J_\alpha(x)$ are solutions of Bessel's differential equation for $\alpha\in\mathbb{C}$ defined by the following equation where $\Gamma$ is the gamma function.
\[J_{\alpha}(x) = \sum_{n=0}^\infty\frac{(-1)^n}{n!\Gamma(n+\alpha+1)}\left(\frac{x}{2}\right)^{2n+\alpha}\]

Definition 2.7.6  The Bessel functions of the second kind or Neumann functions denoted $N_\alpha(x)$ are solutions of Bessel's differential equation for $\alpha\in\mathbb{C}$ defined by the following equation where $J_\alpha$ are the Bessel functions of the first kind.
\[N_\alpha(x) = \frac{J_\alpha(x)\cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}\]

Definition 2.7.7  The Bessel functions of the third kind or Hankel functions denoted $H_\alpha^{(1)}(x),H_\alpha^{(2)}(x)$ are solutions of Bessel's differential equation for $\alpha\in\mathbb{C}$ defined by the following equation where $J_\alpha$ and $N_{\alpha}$ are Bessel functions of the first and second kind respectively.
\[H_\alpha^{(1)}(x) = J_\alpha(x) + i N_\alpha(x)\]\[H_\alpha^{(2)}(x) = J_\alpha(x) - i N_\alpha(x)\]

2.8 Method of Relaxation0123

Definition 2.8.1  The method of relaxation is a numerical method for solving the Laplace equation for a finite region of space with known boundary conditions iteratively. This method computes the potential $V(\vec{r})$ at a finite number of point by iteratively setting each point to the average of the points around it.

File 2.8.2  Relaxation.PNG

2.9 Finite Element MethodZ3RE

Definition 2.9.1  The finite element method is a computational method for calculating an approximation of the solution to a differential equation with a finite mesh to represent the geometry of the system.

2.10 Green's Function TheoryRP3A

Definition 2.10.1  A linear differential operator is a functional operator involving differentiation that is linear.

Definition 2.10.2  The Green's function denoted $G$ for a linear operator $\mathcal{L}$ is the solution to the following equation where $\delta$ is the Dirac delta.
\[\mathcal{L}G(\vec{r},\vec{r}')=-4\pi\delta(\vec{r}-\vec{r}')\]

Corollary 2.10.3  Let $G(\vec{r},\vec{r}')$ be a Green's function for a linear differential operator $\mathcal{L}$ and $F(\vec{r},\vec{r}')$ be any function such that $\mathcal{L}F=0$, then the function $G'(\vec{r},\vec{r}')$ defined below is also a Green's function for $\mathcal{L}$.
\[G'(\vec{r},\vec{r}') = G(\vec{r},\vec{r}') + F(\vec{r},\vec{r}')\]

Definition 2.10.4  The differential form for Green's function theory is the following differential form solving for $u(\vec{r})$ where $\mathcal{L}$ is a linear differential operator and $f(\vec{r})$ is an arbitrary distribution.
\[\mathcal{L}u(\vec{r})=f(\vec{r})\]

Theorem 2.10.5  The solution to a differential equation of the form $\mathcal{L}u(\vec{r})=f(\vec{r})$ is the integral of the Green's function $G$ of the linear differential operator $\mathcal{L}$ and the distribution $f(\vec{r})$.
\[u(\vec{r}) = -\frac{1}{4\pi}\int{ f(\vec{r}')G(\vec{r},\vec{r}') d^3r'}\]

Result 2.10.6  A Green's function for the Laplace operator $G(\vec{r},\vec{r}')$ is the following function. $\newcommand\abs[1]{\left|#1\right|}$
\[G(\vec{r},\vec{r}') = \frac{1}{\abs{\vec{r}-\vec{r}'}}\]

Result 2.10.7  The electric potential of a charge density can be obtained by direct integration.$\newcommand\abs[1]{\left|#1\right|}$
\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int{ \frac{\rho(\vec{r}')}{\abs{\vec{r}-\vec{r}'}} d^3r'}\]

Theorem 2.10.8  The solution to a differential equation of the form $\mathcal{L}u(\vec{r}) = f(\vec{r})$ in the volume $V$ with boundary conditions on surface $S$ is the sum of the following integrals of a Green's function $G$ of the linear differential operator $\mathcal{L}$ and the arbitrary function $f(\vec{r})$, where $\hat{n}$ is the normal vector of the surface $S$ pointing outward from the volume $V$.
\[u(\vec{r}) = -\frac{1}{4\pi}\int_{V}{ f(\vec{r}')G(\vec{r},\vec{r}') d^3r'} + \frac{1}{4\pi}\oint_S\left[\frac{\partial u(\vec{r}')}{\partial n'}\right]G(\vec{r},\vec{r}')d^2r' - \frac{1}{4\pi}\oint_S\left[u(\vec{r}')\right]\frac{\partial G(\vec{r},\vec{r}')}{\partial n'}d^2r'\]

Result 2.10.9  The electric potential of the charge distribution with boundary conditions can be obtain by direct integration with an appropriate choice of Green's function $G$ for the Laplace operator $\nabla^2$. Let $A$ be the volume of interest, $S$ be the boundary surface of $A$ and $\rho(\vec{r})$ be the charge density inside the volume $A$. The electric potential is determined by the following integrals of Green's functions.
\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int_{A}{ \rho(\vec{r}')G(\vec{r},\vec{r}') d^3r'} + \frac{1}{4\pi}\oint_S\left[\frac{\partial V(\vec{r}')}{\partial n'}\right]G(\vec{r},\vec{r}')d^2r' - \frac{1}{4\pi}\oint_S\left[V(\vec{r}')\right]\frac{\partial G(\vec{r},\vec{r}')}{\partial n'}d^2r'\]

Definition 2.10.10  The Dirichlet condition is the boundary condition where the potential $V(\vec{r})$ on the bounding surface is known.

Definition 2.10.11  The Neumann condition is the boundary condition where the normal derivative of the potential $\frac{\partial V(\vec{r})}{\partial n}$ on the bounding surface is known.

Result 2.10.12  The electric potential $V(\vec{r})$ caused by a sphere with known potential can be calculated with integrals of the following Green's Function $G(\vec{r},\vec{r}')$ for the volume $A$ inside or outside a sphere of radius $R$ centered at the origin with bounding surface $S$, where $\hat{n}$ is the normal vector of the surface $S$ pointing outward from the volume $V$.
\[G(\vec{r},\vec{r}') = \frac{1}{\abs{\vec{r}-\vec{r}'}} - \frac{\frac{R}{r'}}{\abs{\vec{r}-\frac{R^2}{r'^2}\vec{r}'}}\]\[\left.\frac{\partial G(\vec{r},\vec{r}')}{\partial n'}\right|_{r'=R} = \frac{-\abs{r^2 - R^2}}{R\left(r^2+R^2-2Rr\left[\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\phi-\phi')\right]\right)^{3/2}}\]\[V(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int_{A}{ \rho(\vec{r}')G(\vec{r},\vec{r}') d^3r'} - \frac{1}{4\pi}\oint_S\left[V(\vec{r}')\right]\frac{\partial G(\vec{r},\vec{r}')}{\partial n'}d^2r'\]

3 Magnetostatics49W1

3.1 Magnetic FieldZEFD

Definition 3.1.1  The magnetic field or magnetic flux density denoted $\vec{B}$ is the vector field that describes the force per length of current in a region of space. This is the actual magnetic field at a point in space.

Definition 3.1.2  The magnetic field strength denoted $\vec{H}$ is the vector that described the external contribution to the magnetic field in a material not intrinsic to the material's magnetization $\vec{M}$, where $\mu_0$ is the vacuum permeability.
\[\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}\]

Definition 3.1.3  A surface current density denoted $\vec{K}(\vec{r})$ is a vector field describing the density of current flowing on a surface.

Result 3.1.4  The current $I$ crossing a line $L$ on a surface $S$ with surface current density $\vec{K}(\vec{r})$ is given by the following path integral relative to a unit vector $\hat{n}$ pointing from the origin to each point on the line.
\[I = \int_L (\vec{K}\times\hat{n})\cdot d\vec{L}\]

Definition 3.1.5  A volume current density denoted $\vec{J}(\vec{r})$ is a vector field describing the density of current flowing at a particular point in space.

Result 3.1.6  The current $I$ passing through a surface $S$ in a space with volume current density $\vec{J}(\vec{r})$ is given by the following surface integral.
\[I = \int_S \vec{J}\times d\vec{S}\]

Law 3.1.7  The Biot-Savart law states that the magnetic field $\vec{B}$ produced by a current carrying wire, a surface current density $\vec{K}(\vec{r}')$ or a volume current density $\vec{J}(\vec{r})$ in a vacuum is determined by the following integrals.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{B} = \frac{\mu_0I}{4\pi}\int_L\frac{d\vec{r}'\times\left(\vec{r}-\vec{r}'\right)}{\abs{\vec{r}-\vec{r}'}^3}\]\[\vec{B} = \frac{\mu_0}{4\pi}\int_S\frac{\vec{K}(\vec{r}')\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^2\vec{r}'\]\[\vec{B} = \frac{\mu_0}{4\pi}\int_V\frac{\vec{J}(\vec{r}')\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3\vec{r}'\]

Law 3.1.8  Ampere's circuit law states that the loop integral of the magnetic field along a loop $C$ is equal to the surface integral of the current flux out of a surface $S$ bounded by $C$. This is a consequence of Stoke's theorem and Maxwell's equations.
\[\oint_C\vec{B}\cdot d\vec{\ell} = \mu_0\int_S\vec{J}\cdot d\vec{S} = \mu_0I_{\text{enclosed}}\]

Result 3.1.9  The force $\vec{F}$ on a current carrying wire with current $I$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following path integral:
\[\vec{F} = I\int{d\vec{\ell}\times \vec{B}(\vec{r}')}\]

Result 3.1.10  The torque $\vec{N}$ on a current density $\vec{J}(\vec{r})$ around a pivot point $\vec{r}_p$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following integral:
\[\vec{N}(\vec{r}_p) = \int \left[\vec{r}' - \vec{r}_p'\right]\times \left[\vec{J}(\vec{r}')\times\vec{B}(\vec{r}')\right]d^3r'\]

3.2 Magnetic ForcesJT8T

Law 3.2.1  The Lorentz force law states that the force $\vec{F}$ on a charged particle with charge $q$ moving at velocity $\vec{v}$ through magnetic field $\vec{B}$ is determined by the following cross product.
\[\vec{F} = q\vec{v}\times\vec{B}\]

Result 3.2.2  The **force $\vec{F}$ on a current density $\vec{J}(\vec{r})$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following integral:
\[\vec{F} = \int{\vec{J}(\vec{r}')\times\vec{B}(\vec{r}')d^3r'}\]

Result 3.2.3  The force $\vec{F}$ on a current carrying wire with current $I$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following path integral:
\[\vec{F} = I\int{d\vec{\ell}\times \vec{B}(\vec{r}')}\]

Result 3.2.4  The torque $\vec{N}$ on a current density $\vec{J}(\vec{r})$ around a pivot point $\vec{r}_p$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following integral:
\[\vec{N}(\vec{r}_p) = \int \left[\vec{r}' - \vec{r}_p'\right]\times \left[\vec{J}(\vec{r}')\times\vec{B}(\vec{r}')\right]d^3r'\]

Result 3.2.5  The torque $\vec{N}$ on a current carrying wire with current $I$ around a pivot point $\vec{r}_p$ in a magnetic field $\vec{B}(\vec{r})$ is given by the following path integral:
\[\vec{N}(\vec{r}_p) = I \int \left[\vec{r}' - \vec{r}_p'\right]\times \left[d\vec{\ell}'\times\vec{B}(\vec{r}')\right]\]

3.3 Magnetic PotentialTD1R

Definition 3.3.1  The magnetic potential denoted $\vec{A}(\vec{r},t)$ is a vector potential such that the curl of the potential is the magnetic field.
\[\vec{B} = \nabla \times \vec{A}\]

Definition 3.3.2  A gauge is a any choice of configuration of the unobservable fields that does not affect the observable fields.

Definition 3.3.3  A gauge transformation is any transformation of the unobservable fields that does not affect the observable fields.

Result 3.3.4  Any arbitrary gauge transformation acting on magnetic potential $\vec{A}(\vec{r},t)$ and electric potential $V(\vec{r},t)$ takes the following form for some scalar field $\Psi(\vec{r},t)$.
\[\vec{A}\to \vec{A} + \nabla\Psi\]\[V \to V - \frac{\partial \Psi}{\partial t}\]

Definition 3.3.5  The Coulomb gauge is the convention that $\nabla\cdot\vec{A} = 0$.

Result 3.3.6  The magnetic potential in the Coulomb gauge can be directly integrated from the volume current density $\vec{J}$ or the magnetic field $\vec{B}$ with one of the following integrals.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{A}(\vec{r}) = \frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}')}{\abs{\vec{r}-\vec{r}'}}d^3r'\]\[\vec{A}(\vec{r}) = \frac{1}{4\pi}\int\frac{\vec{B}(\vec{r}')\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]

3.4 Magnetic Dipole Moment5J8N

Definition 3.4.1  The magnetic dipole moment denoted $\vec{m}$ of a current density $\vec{J}(\vec{r})$ is defined with the following integral.
\[\vec{m} = \frac{1}{2}\int\vec{r}'\times\vec{J}(\vec{r}')d^3\vec{r}'\]

Result 3.4.2  The magnetic potential $\vec{A}$ of a magnetic dipole moment $\vec{m}$ is given by the following formula.
\[\vec{A} = \frac{\mu_0}{4\pi} \frac{\vec{m}\times\vec{r}}{r^3}\]

Result 3.4.3  The magnetic field $\vec{B}$ of a magnetic dipole moment $\vec{m}$ is given by the following formula where $\newcommand\abs[1]{\left|#1\right|}\hat{n} = \frac{\vec{r}-\vec{r}_m}{\abs{\vec{r}-\vec{r}_m}}$ is the unit vector pointing from the position of the magnetic dipole moment $\vec{r}_m$ and the observation point $\vec{r}$.
\[\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi}\left[\frac{3\hat{n}\left(\hat{n}\cdot\vec{m}\right) - \vec{m}}{\abs{\vec{r}-\vec{r}_m}^3} + \frac{8\pi}{3}\vec{m}\delta^3(\vec{r}-\vec{r}_m)\right]\]

Result 3.4.4  The force $F$ on a magnetic dipole moment $\vec{m}$ in a magnetic field $\vec{B}$ is given by the following dot product gradient.
\[\vec{F} = \nabla\left( \vec{m}\cdot\vec{B} \right)\]

Result 3.4.5  The torque $\vec{\tau}$ on a magnetic dipole moment $\vec{m}$ in a magnetic field $\vec{B}$ is given by the following cross product.
\[\vec{\tau} = \vec{m}\times\vec{B}\]

3.5 Magnetic Quadrupole Field1FCD

Definition 3.5.1  The magnetic quadrupole field is the magnetic field produced by four infinitely lone wire run parallel to the z-axis with currents $-I\hat{z}$ at $(0,\pm R,0)$ and $+I\hat{z}$ at $(\pm R,0,0)$.
\[\vec{B} = -\frac{2\mu_0 I}{\pi R} \left[\frac{y\hat{x}+x\hat{y}}{R}\right]\]

4 Special RelativityZNZJ

4.1 Lorentz TransformationsJJJW

Definition 4.1.1  An inertial reference frame is a coordinate system that moves with a constant rectilinear velocity.

Definition 4.1.2  The simultaneity beta denoted $\vec{\beta}$ is the vector $\vec{\beta} = \frac{\vec{v}}{c}$ where $v$ is the velocity of a reference frame and $c$ is the speed of light.

Definition 4.1.3  The simultaneity gamma denoted $\gamma$ is the scalar $\gamma = \left(1-\beta^2\right)^{-1/2}$, where $\newcommand\abs[1]{\left|#1\right|}\beta=\abs{\vec{\beta}}$ is simultaneity beta.

Definition 4.1.4  A Lorentz transformation is a transformation $\Lambda:\mathbb{R}^{1,3}\to\mathbb{R}^{1,3}$ the preserves the scalar $(A_0-B_0)^2-(A_1-B_1)^2-(A_2-B_2)^2-(A_3-B_3)^2$ for all $A,B\in\mathbb{R}^{1,3}$.

Definition 4.1.5  The Poincaré group or the inhomogeneous Lorentz group is the group of all Lorentz transformations such as translations, rotations and boosts.

Definition 4.1.6  A homogeneous Lorentz transformation is a transformation $\Lambda:\mathbb{R}^{1,3}\to\mathbb{R}^{1,3}$ that preserves the scalar $A_0^2-A_1^2-A_2^2-A_3^2$ for all $A\in\mathbb{R}^{1,3}$.

Definition 4.1.7  The homogeneous Lorentz group or sometimes simply the Lorentz group is the group of all homogeneous Lorentz transformations. This only includes linear transformations such as rotations and boosts, but not non-linear transformations such as translations.

Proposition 4.1.8  Homogeneous Lorentz transformations are Lorentz transformations.

Definition 4.1.9  The Lortenz boost $A\mapsto A'$ is a homogeneous Lorentz transformation that transforms any four vector $A\in\mathbb{R}^{1,3}$ to a four vector in a reference frame moving with relative velocity $\vec{v}$ and simultaneity beta $\vec\beta = \frac{\vec{v}}{c}$.
\[A_0' = \gamma(A_0 - \vec{\beta}\cdot\vec{A})\]\[\vec{A}' = \vec{A}+\frac{(\gamma - 1)}{\beta^2}(\vec{\beta}\cdot\vec{A})\vec{\beta}-\gamma\vec{\beta}A_0\]

Result 4.1.10  The Lorentz boost $A\mapsto A'$ into a frame with relative velocity $\vec{v}$ and simultaneity beta $\vec\beta = \frac{\vec{v}}{c}$ can be written in terms of the component $A_\parallel$ of $A$ parallel to $\vec{\beta}$ and the component $A_\perp$ of $A$ perpendicular to $\vec{\beta}$ for $\gamma = \left(1-\beta^2\right)^{-1/2}$.
\[A'_0 = \gamma(A_0 - \vec{\beta}\cdot\vec{A})\]\[A'_\parallel = \gamma(A_\parallel - \beta A_0)\]\[\vec{A}'_{\perp} = \vec{A}'_\perp\]

Result 4.1.11  The following Lorentz boost matrix $\Lambda(\vec{\beta})$ can be used to apply a Lorentz boost $A\mapsto A'=\Lambda(\vec{\beta}) A$ to a four vector $A\in\mathbb{R}^{1,3}$ into a reference frame with relative velocity $\vec{v}$ and simultaneity beta $\vec\beta = \frac{\vec{v}}{c}$.
\[\Lambda(\vec{\beta}) = \begin{pmatrix} \gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z \\
-\gamma\beta_x & 1+\frac{(\gamma-1)\beta_x^2}{\beta^2} & \frac{(\gamma-1\beta_x\beta_y)}{\beta^2} & \frac{(\gamma-1\beta_x\beta_z)}{\beta^2}\\
-\gamma\beta_y & \frac{(\gamma-1\beta_x\beta_y)}{\beta^2} & 1+\frac{(\gamma-1)\beta_y^2}{\beta^2} & \frac{(\gamma-1\beta_y\beta_z)}{\beta^2}\\
-\gamma\beta_z & \frac{(\gamma-1\beta_x\beta_z)}{\beta^2} & \frac{(\gamma-1\beta_y\beta_z)}{\beta^2} & 1+\frac{(\gamma-1)\beta_z^2}{\beta^2} \end{pmatrix}\]

4.2 Four VectorsW147

Definition 4.2.1  A four vector is a vector $A=(A_0,\vec{A})\in\mathbb{R}^{1,3}$ of a time-like scalar component $A_0$ and a space-like vector component $\vec{A}=(A_1,A_2,A_3)$ where the quantity $A_0^2 -A_1^2 -A_2^2 -A_3^2$ is invariant for all reference frames and homogeneous Lorentz transformations.

Definition 4.2.2  The flat negative trace metric tensor is the metric tensor that describes flat space for special relativity with the negative trace convention.
\[\eta = \begin{pmatrix}1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1\end{pmatrix}\]

Definition 4.2.3  The four vector dot product for two four vectors $A,B\in\mathbb{R}^{1,3}$ is defined in terms of a metric tensor $\eta$.
\[A\cdot B = A^\top\eta B\]

Definition 4.2.4  The four vector norm $\newcommand\abs[1]{\left|#1\right|}\abs{A}$ of a four vector $A\in\mathbb{R}^{1,3}$ is defined $\abs{A}=\sqrt{A\cdot A}$.

Result 4.2.5  The dot product of any two four vectors is invariant under Lorentz transformation.

Definition 4.2.6  The 4-position is the four vector $(ct,\vec{r})$ with corresponding invariant spacetime interval $s^2=(ct)^2 - \abs{\vec{r}}^2$ where $c$ is the speed of light, $t$ is time and $\vec{r}$ is the position of a space-time coordinate.

Definition 4.2.7  The 4-velocity is the four vector $(\gamma_vc,\gamma_v\vec{v})$ with corresponding invariant $(\gamma_vc)^2 - \gamma_v^2\abs{\vec{v}}^2$, where $\gamma_v=\left(1-v^2/c^2\right)^{-1/2}$, $c$ is the speed of light and $\vec{v}$ is the velocity.

Result 4.2.8  The velocity addition law describes how to combine the velocities of two objects in a way that is consistent with relativity.

Definition 4.2.9  The 4-momentum is the four vector $(E/c,\vec{p})$ with corresponding invariant $(E/c)^2 - \abs{\vec{p}}^2 = m^2c^2$ where $c$ is the speed of light, $m$ is the rest mass, $E$ is the energy and $\vec{p}$ is the momentum.

Definition 4.2.10  The 4-current density is the four vector $\left(c\rho,\vec{J}\right)$ with corresponding invariant $(c\rho)^2 - \abs{\vec{J}}^2$ where $c$ is the speed of light, $\rho$ is the charge density and $\vec{J}$ is the current density.

Definition 4.2.11  The 4-wave vector is the four vector $(\omega/c,\vec{k})$ with corresponding invariant $(\omega/c)^2 - \abs{\vec{k}}^2 = (mc/\hbar)^2$ where $c$ is the speed of light, $\omega$ is the frequency in radians per second, $\vec{k}$ is the wave vector and $m$ is the rest mass.

Result 4.2.12  The phase of a wave $\phi = \omega t - \vec{k}\cdot\vec{r}$ is invariant under Lorentz transformation.
\[(ct,\vec{r})\cdot(\omega/c,\vec{k}) = \omega-\vec{k}\cdot\vec{r} = \phi\]

Table 4.2.13  Four vectors and their corresponding invariants.$\newcommand\abs[1]{\left|#1\right|}$

4.3 Einstein NotationM460

Definition 4.3.1  Einstein notation or Einstein summation notation is a notational convention that used subscripts and superscripts to simplify summation notation over the implied range of that index. Paired indexes are summed over and unpaired indexes represent the components of the vector or tensor.
\[c_ix^i = \sum_{i=0}^3 c_ix^i = c_0x^0 + c_1x^1 + c_2x^2 + c_3x^3\]

Definition 4.3.2  A contravarient vector is a four vector written in Einstein notation with the index in the superscript.
\[A^\mu = (A_0,\vec{A})\]

Definition 4.3.3  A covarient vector is a four vector written in Einstein notation with the index in the subscript.
\[A_\mu = \eta_{\mu\nu}A^\nu = \sum_{\nu=0}^3\eta_{\mu\nu}A^\nu = (A_0,-\vec{A})\]

Definition 4.3.4  The scalar product of two four vectors $a,b\in\mathbb{R}^{1,3}$ is simply the four vector dot product.
\[a\cdot b = a_\mu b^\mu = \eta_{\mu\nu}a^\nu b^\mu = \sum_{\mu =0}^3\sum_{\nu=0}^3\eta_{\mu\nu}a^\nu b^\mu = a_0b_0 - \vec{a}\cdot\vec{b}\]

4.4 Field Strength Tensor65TE

Definition 4.4.1  The field strength tensor $F$ is a 4-tensor that contains both the electric field and the magnetic field at a particular point in spacetime.
\[F = \begin{pmatrix} 0 & -E_x & -E_y & -E_z\\ +E_x & 0 & -B_z & +B_y \\ +E_y & +B_z & 0 & -B_x \\ +E_z & -B_y & +B_x & 0\end{pmatrix}\]

Result 4.4.2  The Lorentz boost of a 4-tensor can be computed by applying the Lorentz boost matrix $\Lambda(\vec\beta)$ to a 4-tensor $T$.
\[T' = \Lambda(\vec{\beta})T\Lambda^\top\]

Result 4.4.3  The Lorentz boost of electric and magnetic fields into a frame with relative velocity $\vec{v}$ and simultaneity beta $\vec\beta = \frac{\vec{v}}{c}$ can be calculated with the following formulas.
\[\vec{E}' = \gamma\left(\vec{E} + \vec\beta\times\vec{B}\right) - \frac{\gamma - 1}{\beta^2}\left(\vec\beta \cdot\vec{E}\right)\vec\beta\]\[\vec{B}'=\gamma\left(\vec{B}-\vec\beta\times\vec{E}\right)-\frac{\gamma-1}{\beta^2}\left(\vec\beta\cdot\vec{B}\right)\vec\beta\]

Result 4.4.4  The Lorentz boost of electric and magnetic fields into a frame with relative velocity $\vec{v}$ and simultaneity beta $\vec\beta = \frac{\vec{v}}{c}$ can be written in terms of the components $E_\parallel,B_\parallel$ of $E,B$ parallel to $\vec{\beta}$ and the components $E_\perp,B_\perp$ of $E,B$ perpendicular to $\vec{\beta}$.
\[\vec{E}'_\parallel = \vec{E}_\parallel\]\[\vec{E}'_\perp = \gamma\left(\vec{E}_\perp + \vec\beta\times\vec{B}_\perp\right)\]\[\vec{B}'_\parallel = \vec{B}_\parallel\]\[\vec{B}'_\perp = \gamma\left(\vec{B}_\perp - \vec\beta\times\vec{E}_\perp\right)\]

Result 4.4.5  The relativistic motion in static electromagnetic fields for a charged particle with rest mass $m$ and momentum $\vec{p}$ in electric and magnetic fields $\vec{E}$ and $\vec{B}$ the motion of this particles can be described by the following differential equation.
\[\frac{\partial \vec{p}}{\partial t} = \frac{\partial (\gamma mc\vec{\beta})}{\partial t} = q\left[ \vec{E} + \vec\beta\times\vec{B} \right]\]

5 Electrodynamics89DF

5.1 Slowly Varying FieldsHE45

Definition 5.1.1  The electromotive force $\varepsilon_{EMF}$ or electromotive voltage is the force per unit charge integrated along a loop moving at velocity $\vec{v}$.
\[\varepsilon_{EMF} = \oint\frac{F}{q}\cdot d\vec{\ell}=\oint\left[ \vec{E} + \vec{v}\times\vec{B} \right]\cdot d\vec{\ell}\]

Definition 5.1.2  The magnetic flux $\Phi_B$ is the integral of the magnetic field $\vec{B}$ dotted with the normal vector for a surface $S$.
\[\Phi_B = \int_{S}\vec{B}\cdot d\vec{S}\]

Law 5.1.3  Faraday's law of induction states that the electromotive force around a loop of wire $\varepsilon_{EMF}$ is equal to the negative time derivative of a slowly varying magnetic flux $\Phi$ through a surface $S$ bounded by the loop.
\[\varepsilon_{EMF} = -\frac{\partial \Phi_B}{\partial dt}\]

Result 5.1.4  The time derivative of the magnetic flux can be rewritten as the following two integrals of the surface $S$ and boundary path $P$.
\[-\frac{\partial \Phi_B}{\partial dt} = - \int_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{S} - \oint_P(\vec{v}\times \vec{B})\cdot d\vec{P}\]

Law 5.1.5  The expanded Faraday's law of induction states that transformer electromotive force is equal to the time derivative of a slowly varying transformer magnetic flux.
\[\oint\vec{E}\cdot d\vec{\ell} = - \int_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{S}\]

Definition 5.1.6  The displacement current density $\vec{J}_D$ is the effective current density of a changing electric field.
\[\vec{J}_D = \varepsilon_0\frac{\partial \vec{E}}{\partial t}\]

Result 5.1.7  The magnetic field with displacement current is the result of Stokes theorem to Maxwell's equations and displacement current density for slowly varying fields.
\[\oint\vec{B}\cdot d\vec{\ell} = \mu_0 \int_S \left(\vec{J} + \vec{J}_D\right)\cdot d\vec{S}\]\[\oint\vec{B}\cdot d\vec{\ell} =\mu_0 \int_S \vec{J} \cdot d\vec{S} + \mu_0\varepsilon_0 \int_S \frac{\partial \vec{E}}{\partial t}\cdot d\vec{S}\]

5.2 Retarded Time0JDF

Definition 5.2.1  The retarded time $t'$ is the time in the past that position $r'$ appears at when observed from position $r$ and time $t\newcommand\abs[1]{\left|#1\right|}$ due to speed of light $c$ delay.
\[t' = t - \frac{\abs{\vec{r}-\vec{r}'}}{c}\]

Law 5.2.2  The wave equations for electric potential $V$ and magnetic potential $\vec{A}$ describes how the potentials behave with time varying charge density $\rho$ and current density $\vec{J}$.
\[\nabla^2V - \mu_0\varepsilon_0\frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0}\]\[\nabla^2\vec{A}-\mu_0\varepsilon_0\frac{\partial^2 \vec{A}}{\partial t^2} = -\mu_0\vec{J}\]\[\vec{\nabla}\cdot\vec{A} +\mu_0\varepsilon_0\frac{\partial V}{\partial t} = 0\]

Result 5.2.3  A Time dependent Green's function for the Laplace operator $G(\vec{r},\vec{r}',t,t')$ is the following function. $\newcommand\abs[1]{\left|#1\right|}$
\[G(\vec{r},\vec{r}',t,t') = \frac{\delta\left(t' - \left[ t - \frac{\abs{\vec{r}-\vec{r}'}}{c} \right]\right)}{\abs{\vec{r}-\vec{r}'}}\]

Result 5.2.4  The electric potential of a time dependent charge density can be obtained by direct integration.$\newcommand\abs[1]{\left|#1\right|}$
\[V(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int{ \frac{\rho(\vec{r}',t')}{\abs{\vec{r}-\vec{r}'}} d^3r'}\]\[V(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int{ \frac{\rho(\vec{r}',t - \frac{\abs{\vec{r}-\vec{r}'}}{c})}{\abs{\vec{r}-\vec{r}'}} d^3r'}\]

5.3 Jefimenko's Equations0EAK

Definition 5.3.1  Jefimenko's equations are a series of integrals that can be used to evaluate the exact electric and magnetic field produced by a time varying charge density $\rho(\vec{r},t)$ and current density $\vec{J}(\vec{r},t)$.

Result 5.3.2  Jefimenko's equation for electric fields describes the exact electric field $\vec{E}(\vec{r},t)$ produced by a time varying charge density $\rho(\vec{r},t)$ and current density $\vec{J}(\vec{r},t)$ where $t'$ is the retarded time.
\[\vec{E}(\vec{r},t) = \vec{E}_{Coulomb}(\vec{r},t) + \vec{E}_{Charge}(\vec{r},t)+\vec{E}_{Current}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\]\[\vec{E}_{Coulomb}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int \rho(\vec{r}',t')\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]\[\vec{E}_{Charge}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left[ \frac{1}{c}\frac{\partial \rho(\vec{r}',t')}{\partial t'} \right] \frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^2}d^3r'\]\[\vec{E}_{Current}(\vec{r},t)=-\frac{1}{4\pi\varepsilon_0}\int\left[\frac{1}{c^2}\frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\right]\frac{1}{\abs{\vec{r}-\vec{r}'}}d^3r'\]

Result 5.3.3  Alternate Jefimenko's equation for electric fields describes the exact electric field $\vec{E}(\vec{r},t)$ produced by a time varying charge density $\rho(\vec{r},t)$ and current density $\vec{J}(\vec{r},t)$. When $\nabla\cdot\vec{J} = 0 $, $\vec{E}_{Currrent1} = 0$ where $t'$ is the retarded time.
\[\vec{E}(\vec{r},t) = \vec{E}_{Coulomb}(\vec{r},t)+\vec{E}_{Current1}(\vec{r},t)+\vec{E}_{Current2}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\]\[\vec{E}_{Coulomb}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int \rho(\vec{r}',t')\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]\[\vec{E}_{Current1}(\vec{r},t)=\frac{1}{4\pi\varepsilon_0c}\int\frac{\left[\vec{J}(\vec{r}',t')\cdot(\vec{r}-\vec{r}')\right](\vec{r}-\vec{r}')+\left[\vec{J}(\vec{r}',t')\times(\vec{r}-\vec{r}')\right]\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^4}d^3r'\]\[\vec{E}_{Current2}(\vec{r},t)=\frac{1}{4\pi\varepsilon_0c^2}\int\frac{\left[\frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\times(\vec{r}-\vec{r}')\right]\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]

Result 5.3.4  Jefimenko's equation for magnetic fields describes the exact magnetic field $\vec{B}(\vec{r},t)$ produced by a time varying charge density $\rho(\vec{r},t)$ and current density $\vec{J}(\vec{r},t)$ where $t'$ is the retarded time.
\[\vec{B}(\vec{r},t) = \vec{B}_{BiotSavart}(\vec{r},t) + \vec{B}_{Current}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\]\[\vec{B}_{BiotSavart}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\vec{J}(\vec{r}',t')\times\left[\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}\right]d^3r'\]\[\vec{B}_{Current}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\left[ \frac{1}{c} \frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\right]\times\left[\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^2}\right]d^3r'\]

Result 5.3.5  The quasistatic expanded Jefimenko's equation is an expansion of Jefimenko's equations that approximates the electric field $\vec{E}(\vec{r},t)$ and magnetic field $\vec{B}(\vec{r},t)$ produced by a time varying charge density $\rho(\vec{r},t)$ and current density $\vec{J}(\vec{r},t)$ where $t'$ is the retarded time.$\newcommand\abs[1]{\left|#1\right|}$
\[\vec{E}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left(\rho(\vec{r}',t) - \frac{\abs{\vec{r}-\vec{r}'}^2}{2c^2}\frac{\partial^2\rho(\vec{r}',t)}{\partial t^2} + \cdots\right)\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]\[+\frac{\mu_0}{4\pi}\int\frac{\partial \vec{J}(\vec{r}',t)}{\partial t}\frac{1}{\abs{\vec{r}-\vec{r}'}}+\cdots d^3r'\]\[\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\left[\vec{J}(\vec{r}',t)-\frac{\abs{\vec{r}-\vec{r}'}^2}{2c^2}\frac{\partial^2\vec{J}(\vec{r}',t)}{\partial t^2}\right]\times\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\]

5.4 Electromagnetic Energy and Momentum6F89

Definition 5.4.1  The electromagnetic energy density $u$ is the energy density of the electromagnetic fields at a particular point defined in terms of the electric field $\vec{E}$, the electric displacement field $\vec{D}$, the magnetic field $\vec{B}$ and the magnetic field strength $\vec{H}$.
\[u = \frac{\vec{E}\cdot \vec{D} + \vec{B}\cdot\vec{H}}{2}\]

Result 5.4.2  In a electromagnetic energy density in a vacuum $u$ can be written in terms of the electric field $\vec{E}$ and magnetic field $\vec{B}$.
\[u = \frac{\varepsilon_0\vec{E}\cdot\vec{E} + \vec{B}\cdot\vec{B}/\mu_0}{2}\]

Definition 5.4.3  The Poynting vector $\vec{S}$ is the energy flux (energy per unit area per unit time) of electromagnetic fields at a particular point defined in terms of the electric field $\vec{E}$ and magnetic field strength $\vec{H}$.
\[\vec{S} = \vec{E}\times\vec{H}\]

Result 5.4.4  The Poynting vector in a vacuum $\vec{S}$ can be written in terms of the electric field $\vec{E}$ and magnetic field $\vec{B}$.
\[\vec{S} = \frac{\vec{E}\times\vec{B}}{\mu_0}\]

Law 5.4.5  The electromagnetic conservation of energy law states that energy is conserved by electromagnetic fields.
\[\frac{\partial u}{\partial t} + \nabla\cdot\vec{S} = -\vec{J}\cdot\vec{E}\]\[\vec{J}\cdot\vec{E} + \frac{\partial u}{\partial t} = -\nabla\cdot\vec{S}\]

Definition 5.4.6  The electromagnetic momentum density $\vec{g}$ is the momentum density of the electromagnetic fields at a particular point defined in terms of the electric field $\vec{E}$ and magnetic field strength $\vec{H}$.
\[\vec{g} = \frac{\vec{E}\times\vec{H}}{c^2} = \frac{\vec{S}}{c^2}\]

Definition 5.4.7  The Maxwell stress tensor or momentum flux tensor $T_{ij}$ is the 3 by 3 matrix describing the momentum flux (momentum per unit area per unit time) that is the change in each component of momentum in each direction defined by the the electric field $\vec{E}$ and magnetic field $\vec{B}$ in a vacuum.
\[T_{ij} = \varepsilon_0\left[ E_iE_j + c^2 B_iB_j - \frac{1}{2}\left(\vec{E}\cdot\vec{E} + c^2\vec{B}\cdot\vec{B}\right)\delta_{i,j}\right]\]

Law 5.4.8  The electromagnetic conservation of momentum law states that momentum is conserved by electromagnetic fields.
\[\frac{d\vec{p}_{mech}}{dt} = q(\vec{E} + \vec{v}\times\vec{B})\]\[\vec{p}_{field} = \varepsilon_0\int_V\vec{E}\times\vec{B} dV = \int_V \vec{g} dV\]\[\left[ \frac{d\vec{p}_{mech}}{dt} + \frac{d \vec{p}_{field}}{dt}\right]\cdot \hat{k} = \oint_S\sum_jT_{k,j}n_j dS\]
where $k\in\{x,y,z\}$ and $\hat{n}$ is the vector pointing out of the surface $S$ that bounds the volume $V$.

Definition 5.4.9  The Maxwell stress 4-tensor $\Omega^{\alpha,\beta}$ is the 4-tensor that can be used to describe the energy and momentum of electromagnetic fields in different reference frames.
\[\Omega = \begin{pmatrix} u & c\vec{g} \\ c\vec{g} & -T \end{pmatrix} =
\begin{pmatrix}
u & cg_x & cg_y & cg_z\\
cg_x & -T_{xx} & -T_{xy} & T_{xz}\\
cg_y & -T_{yx} & - T_{yy} & - T_{yz}\\
cg_z & -T_{zx} & -T_{zy} & -T_{zz}
\end{pmatrix}\]

Result 5.4.10  The Lorentz boost of a 4-tensor can be computed by applying the Lorentz boost matrix $\Lambda(\vec\beta)$ to a 4-tensor $T$.
\[T' = \Lambda(\vec{\beta})T\Lambda^\top\]

5.5 Multipole RadiationKT3A

Result 5.5.1  The power radiated from a rotating multipole $P_{radiated}$ can be written in terms of the oscillation frequency and the multipole expansion $\vec{p},\vec{m},Q_{jk},\cdots\newcommand\abs[1]{\left|#1\right|}$ of the charge distribution.
\[P_{radiated} = \frac{Z_0\omega^4}{12\pi c^4}\left[ \abs{\vec{p}}^2 + \abs{\frac{\vec{m}}{c}}^2 + \frac{\omega^2}{120 c^2}\sum_{jk}\abs{Q_{jk}}^2 + \cdots \right]\]

Result 5.5.2  The solid angle distribution of multipole radiation describes the angular distribution of the power radiated by a rotating multipole.
\[\frac{\partial P}{\partial \Omega} \propto \omega^4 q^2 (r')^2\sin^2(\theta)\]\[\frac{\partial P}{\partial \Omega} \propto \omega^4\left[ \frac{I}{c}(r') \right]^2(r')^2\sin^2(\theta)\]\[\frac{\partial P}{\partial \Omega} \propto \omega^4\left[ \omega\frac{q}{c}(r') \right]^2(r')^2\sin^2(\theta)\cos^2(\theta)\]

Definition 5.5.3  The vector spherical harmonics $\vec{X}_{\ell,m}(\theta,\phi)$ are generalizations of spherical harmonics for vectors fields.
\[\vec{X}_{\ell,m}(\theta,\phi) = \frac{-i(\vec{r}\times\nabla)Y_{\ell,m(\theta,\phi)}}{\sqrt{\ell(\ell + 1)}}\]

Result 5.5.4  The general electromagnetic multipole radiation generated by electric multiple moments $a_M^{\ell,m}$ and magnetic multipole moments $a_E^{\ell,m}$ oscillating at frequency $\omega$ can be written in terms of Vector Spherical Harmonics $\vec{X}_{\ell,m}(\theta,\phi)$.
\[\vec{E} = \text{Real}\left( \sqrt{\frac{\mu_0}{\varepsilon_0}} \sum_{\ell,m}\left[ a_E^{\ell,m}\nabla\times\left( \frac{i}{k}f_\ell(kr)\vec{X}_{\ell,m} \right) + a_M^{\ell,m}g_\ell(kr) \vec{X}_{\ell,m} \right] e^{-i\omega t} \right)\]\[\vec{B} = \text{Real}\left( \mu_0\sum_{\ell,m}\left[ -a_E^{\ell,m}\nabla\times\left( \frac{i}{k}g_\ell(kr)\vec{X}_{\ell,m} \right) + a_M^{\ell,m}f_\ell(kr) \vec{X}_{\ell,m} \right] e^{-i\omega t} \right)\]\[f_\ell(kr) = A_\ell^{(1)}H_\ell^{(1)}(kr) + A_\ell^{(2)}H_\ell^{(2)}(kr)\]\[g_\ell(kr) = B_\ell^{(1)}H_\ell^{(1)}(kr) + B_\ell^{(2)}H_\ell^{(2)}(kr)\]\[k = \frac{\omega}{c}\]
where $A_\ell^{(1)}, A_{\ell}^{(2)}, B_\ell^{(1)}, B_\ell^{(2)}$ are coefficients determined by boundary conditions and $H_\ell^{(1)}, H_\ell^{(2)}$ are Hankel functions.

Result 5.5.5  The electric multipole moment $a_E^{\ell,m}$ due to an oscillating charge density $\rho$ oscillating at angular frequency $\omega$ (wavenumber $k=\omega/c$) can be written as integrals of spherical harmonics $Y_{\ell,m}^*$ and Bessel functions $J_{\ell + 1/2}$. The multipole moment can be approximated when the wavelength is much greater that the size of the source.
\[a_E^{\ell,m} = \frac{k^2}{i\sqrt{\ell(\ell + 1)}} \int Y_{\ell,m}^*(\theta,\phi)\left[ c \rho(\vec{r}) \sqrt{\frac{\pi}{2kr}}\frac{\partial \left(r J_{\ell + 1/2}(kr)\right)}{\partial r}\right] d^3r\]\[a_E^{\ell,m}\approx\frac{ck^{\ell+2}}{i(2\ell + 1)!!}\sqrt{\frac{\ell + 1}{\ell}}\int r^\ell Y_{\ell,m}^*(\theta,\phi)\rho(\vec{r})d^3r\]

Result 5.5.6  The multipole moment $a_E^{\ell,m},a_M^{\ell,m}$ due to an oscillating current density $\vec{J}$ oscillating at angular frequency $\omega$ (wavenumber $k=\omega/c$) can be written as integrals of spherical harmonics $Y_{\ell,m}^*$ and Bessel functions $J_{\ell + 1/2}$. The multipole moment can be approximated when the wavelength is much greater that the size of the source.
\[a_E^{\ell,m} = \frac{k^2}{i\sqrt{\ell(\ell + 1)}}\int Y_{\ell,m}^*(\theta,\phi)\left[ ik\vec{r}\cdot\vec{J}(\vec{r})\sqrt{\frac{\pi}{2kr}}J_{\ell + 1/2}(kr) \right] d^3r\]\[a_E^{\ell,m} \approx 0\]\[a_M^{\ell,m} = \frac{k^2}{i\sqrt{\ell(\ell + 1)}}\int Y_{\ell,m}^*(\theta,\phi)\left[ \nabla \cdot \left( \vec{r}\times\vec{J}(\vec{r})\right) \sqrt{\frac{\pi}{2kr}}J_{\ell + 1/2}(kr)\right] d^3r\]\[a_M^{\ell,m}\approx \frac{ik^{\ell+2}}{(2\ell + 1)!!}\sqrt{\frac{\ell}{\ell + 1}}\int (r)^\ell Y_{\ell,m}^*(\theta,\phi)\nabla\cdot\left( \vec{J}\times\vec{r} \right) d^3r\]

Result 5.5.7  The multipole moment $a_E^{\ell,m},a_M^{\ell,m}$ due to an oscillating magnetization $\vec{M}$ oscillating at angular frequency $\omega$ (wavenumber $k=\omega/c$) can be written as integrals of spherical harmonics $Y_{\ell,m}^*$ and Bessel functions $J_{\ell + 1/2}$. The multipole moment can be approximated when the wavelength is much greater that the size of the source.
\[a_E^{\ell,m} = \frac{k^2}{i\sqrt{\ell(\ell + 1)}} \int Y_{\ell,m}^*(\theta,\phi)\left[ -ik\nabla\cdot\left( \vec{r}\times\vec{M}(\vec{r}) \right) \sqrt{\frac{\pi}{2kr}}J_{\ell+1/2}(kr) \right] d^3r\]\[a_E^{\ell,m} \approx -\frac{k^{\ell + 2}}{(2\ell + 1)!!}\sqrt{\frac{1}{\ell(\ell + 1)}}\int(r)^\ell Y_{\ell,m}^*(\theta,\phi)\left[ \nabla\cdot(\vec{r}\times\vec{M}(\vec{r})) \right] d^3r\]\[a_M^{\ell,m} = \frac{k^2}{i\sqrt{\ell(\ell+1)}}\int Y_{\ell,m}^*(\theta,\phi)\left[ -k^2\left(\vec{r}\cdot\vec{M}(\vec{r})\right)\sqrt{\frac{\pi}{2kr}}J_{\ell + 1/2}(kr) \right]d^3r\]\[a_M^{\ell,m}\approx -\frac{ik^{\ell+2}}{(2\ell+1)!!}\sqrt{\frac{\ell + 1}{\ell}}\int (r)^\ell Y_{\ell,m}^*(\theta,\phi)\left( \nabla\cdot\vec{M}(\vec{r}) \right)d^3r\]

5.6 Moving ChargesZ3CD

Result 5.6.1  The electromagnetic fields from an accelerating point charge at position $\vec{r}_{charge}$ observed at position $\vec{r}_{obs}$ can be written as the following two terms. The first term is the "velocity field" which dominates in the quasi-static regime and the second term is the "acceleration field" which dominates in the radiation regime.
\[\vec{E} = \frac{q}{4\pi\varepsilon_0}\left[\frac{\beta}{\gamma^2 R^2}\cdot\frac{\hat{R}/\beta - \hat{\beta}}{\left(1-\vec{\beta}\cdot\hat{R}\right)^3}\right]_{ret} + \frac{q}{4\pi\varepsilon_0}\left[\frac{\beta\dot{\beta}}{cR}\cdot\frac{\hat{R}\times\left( \left(\hat{R}/\beta - \hat{\beta}\right)\times\dot{\hat{\beta}} \right)}{\left(1-\vec{\beta}\cdot\hat{R}\right)^3}\right]_{ret}\]\[c\vec{B} = \left[\hat{R}\times\vec{E}\right]_{ret}\]\[\vec{R} = \vec{r}_{obs}-\vec{r}_{charge}\]

Definition 5.6.2  The Larmor power $P_{Larmor}$ is the power radiated by an accelerating particle with acceleration $\dot{\beta}$ in the non-relativistic limit.
\[P_{Larmor} = \frac{q^2 \dot\beta^2}{6\pi\varepsilon_0 c}\]

Result 5.6.3  The relativistic Larmor formula describes the power radiated $P_{rad}\newcommand\abs[1]{\left|#1\right|}$ by a relativistic accelerating particle.
\[P_{rad} = \gamma^6\left[1-\abs{\hat{\beta}\times\hat{\dot{\beta}}}^2\beta^2\right]P_{Larmor} = \gamma^6\left[1-\abs{\hat{\beta}\times\hat{\dot{\beta}}}^2\beta^2\right] \frac{q^2 \dot\beta^2}{6\pi\varepsilon_0 c}\]\[\frac{dP_{rad}}{d\Omega} = \frac{q^2}{16\pi^2\varepsilon_0c}\frac{\abs{\hat{R}\times\left(\left(\hat{R}-\vec{\beta}\right)\times\dot{\vec{\beta}}\right)}^2}{\left(1-\hat{R}\cdot\vec{\beta}\right)^5}\]\[\vec{R} = \vec{r}_{obs}-\vec{r}_{charge}\]

6 AppendixPPMA

6.1 ReferencesN5DH

¹

Electromagnetism in Materials2HPE

1 Introduction30CA

1.1 LogisticsZ5TM

File 1.1.1  syllabus_EM2_2024.pdf

1.2 Notation3D4P

Definition 1.2.1  The Kronecker delta $\delta_{ij} = \left\{\begin{array}{lr}
1, & \text{if } i = j\\
0, & \text{if } i\neq j
\end{array}\right\}$.

Definition 1.2.2  The Levi-Civita symbol $\varepsilon_{a_1,a_2,\dots,a_n} = \left\{\begin{array}{rl}
+1, & \text{if } (a_1,a_2,\dots,a_n)\text{ is an even permutation of }(1,2,\dots,n)\\
-1, & \text{if } (a_1,a_2,\dots,a_n)\text{ is an odd permutation of }(1,2,\dots,n)\\
0, & \text{otherwise}
\end{array}\right\}$

Definition 1.2.3  The dot product denoted $\vec{a}\cdot\vec{b}$ of two vectors $\vec{a}$ and $\vec{b}$ is the sum of the products of there components in any orthonormal basis.
\[\vec{a}\cdot\vec{b} = \sum_{i}a_ib_i\]

Definition 1.2.4  The cross product of two vectors $\vec{a}$ and $\vec{b}$ is defined by the following sums of the products of there components in any orthonormal basis $\vec{e_1},\vec{e_2},\dots,\vec{e_n}$.
\[\vec{a}\times\vec{b} = \sum_i\sum_j\sum_k\varepsilon_{ijk}a_jb_k \vec{e_i}\]

Definition 1.2.5  The gradient denoted $\nabla F$ of a differentiable scalar field $F$ is the vector field defined by the partial derivatives of the scalar field.
\[\nabla F = \frac{\partial F}{\partial x}\hat{x} + \frac{\partial F}{\partial y}\hat{y} + \frac{\partial F}{\partial z}\hat{z}=\left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right)\]\[\nabla F = \frac{\partial F}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial F}{\partial \theta}\hat{\theta} + \frac{1}{r\sin\theta} \frac{\partial F}{\partial \phi}\hat{\phi}\]

Definition 1.2.6  The divergence denoted $\nabla \cdot \vec{F}$ of a differentiable vector field $\vec{F}$ is a scalar field defined by the sum of partial derivatives of the components of the vector field.
\[\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\]\[\nabla \cdot \vec{F} = \frac{1}{r^2}\frac{\partial (r^2F_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta} \left(F_\theta \sin\theta\right) + \frac{1}{\sin\theta}\frac{\partial F_\phi}{\partial \phi}\]\[\nabla \cdot \vec{F} = \frac{1}{s}\frac{\partial (sF_s)}{\partial s} + \frac{1}{s} \frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z}\]

Definition 1.2.7  The curl denoted $\nabla \cross \vec{F}$ of a differentiable vector field $\vec{F}$ is a vector field defined in terms of the following partial derivatives.
\[\nabla \times \vec{F}=\det\begin{pmatrix}
\hat{x} & \hat{y} & \hat{z}\\
\frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\
F_x & F_y & F_z
\end{pmatrix}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\hat{y}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\hat{z}\]\[\nabla \times \vec{F} = \frac{1}{r\sin\theta}\left( \frac{\partial }{\partial\theta}(F_\phi \sin\theta) - \frac{\partial F_\theta}{\partial \phi} \right)\hat{r} + \frac{1}{r}\left( \frac{1}{\sin\theta}\frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}(rF_\phi)\right)\hat{\theta} + \frac{1}{r}\left(\frac{\partial }{\partial r}(rF_\theta)-\frac{\partial F_r}{\partial \theta}\right)\]\[\nabla \times \vec{F} = \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right)\hat{s} + \left( \frac{\partial F_s}{\partial z} - \frac{\partial F_z}{\partial s} \right)\hat{\phi} + \frac{1}{s}\left(\frac{\partial (sF_\phi)}{\partial s} - \frac{\partial F_s}{\partial \phi} \right)\hat{z}\]

Definition 1.2.8  The Laplace denoted $\nabla^2 F$ of a differentiable scalar field $F$ is the divergence of the gradient of $F$.
\[\nabla^2F = \nabla\cdot\nabla F = \frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} + \frac{\partial^2 F}{\partial z^2}\]

Proposition 1.2.9  The curl of a gradient is zero, that is $\nabla\times(\nabla F)=0$ for any scalar field $F$.

Proposition 1.2.10  The divergence of a curl is zero, that is $\nabla\cdot(\nabla \times \vec{F})=0$ for any vector field $\vec{F}$.

Proposition 1.2.11  The curl of curl can be written in terms of gradients and divergences for any vector field $\vec{F}$.
\[\nabla\times\left(\nabla\times\vec{F}\right) = \nabla\left( \nabla\cdot\vec{F} \right) - \nabla\cdot\nabla\vec{F} = \nabla\left( \nabla\cdot\vec{F} \right) - \nabla^2\vec{F}\]

Theorem 1.2.12  The gradient theorem states that gradients are path independent, that is for any differentiable scalar field $F$,
\[\int_\vec{a}^\vec{b}(\nabla F)\cdot d\vec{\ell} = F(\vec{b})-F(\vec{a})\]

Theorem 1.2.13  The divergence theorem or Gauss's theorem states that the surface integral of continuously differentiable vector field $\vec{F}$ dotted with the normal vector $d\vec{S}$ over the piecewise smooth boundary $S$ of a volume $V$ is the volume integral of the divergence of $\vec{F}$ over the volume $V$.
\[\iint_S \vec{F}\cdot d\vec{S} = \iiint_V(\nabla\cdot\vec{F})dV\]

Theorem 1.2.14  The curl theorem or Stokes' theorem states that the loop integral of a continuously differentiable vector field $\vec{F}$ along the boundary path $P$ of a smooth oriented surface $S$ is the surface integral of the curl of $\vec{F}$ over the surface $S$.
\[\int_P\vec{F}\cdot d\vec{P} = \iint_S(\nabla\times\vec{F})\cdot d\vec{S}\]

Definition 1.2.15  The Dirac delta denoted $\delta(x)$ is the notation for a distribution peaked infinitesimally around zero.
\[\delta(x) = \lim_{a\to 0} \frac{1}{|a|\sqrt{\pi}}e^{-(x/a)^2}\]

1.3 Gaussian UnitsCR76

Definition 1.3.2  The SI unit system is the most popular system of units that uses the fundamental units of seconds, meters, kilograms, ampere and Kelvin to derive a system of units to describe the universe.

Definition 1.3.3  A second (s) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

Definition 1.3.4  A meter (m) is the SI unit of distance that is exactly the distance light travels in $1/299792458$ seconds.

Definition 1.3.5  A kilogram (kg) is the SI unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−34}\text{kg } \text{m}^2\text{s}^{−1}$.

Definition 1.3.6  An ampere (A) is the SI unit of current that is exactly the flow of $10^{19}/1.602176634$ protons per second.

Definition 1.3.7  A Kelvin (K) is the SI unit of absolute temperature defined exactly by fixing Boltzmann's constant $k=1.380649\times 10^{-23}\text{kg }\text{m}^2 \text{s}^{-2}\text{K}^{-1}$.

Definition 1.3.8  A Coulomb (C) is the SI unit of charge defined by $\text{C} = \text{A s}$ or exactly $10^{19}/1.602176634$ protons.

Definition 1.3.9  A Newton (N) is the SI unit of force defined by $\text{N} = \text{kg m}/\text{s}^{2}$.

Definition 1.3.10  A Joule (J) is the SI unit of energy defined by $\text{J} = \text{N m} = \text{kg }\text{m}^{2}/\text{s}^{2}$.

Definition 1.3.11  A Watt (w) is the SI unit of power defined by $\text{w} = \text{J}/\text{s}$.

Definition 1.3.12  A Pascal (Pa) is the SI unit of pressure defined by $\text{Pa}=\text{N}/\text{m}^2 = \text{J}/\text{m}^3 = \text{kg }\text{m}^{-1}\text{s}^{-2}$.

Definition 1.3.13  A Volt (V) is the SI unit of electric potential defined by $\text{V} = \text{J}/\text{C} = \text{w}/\text{A} = \text{kg }\text{m}^2\text{s}^{-3}\text{A}^{-1}$.

Definition 1.3.14  A Volt per Meter (V/m) is the SI unit of electric field defined by $\text{V}/\text{m} = \text{N}/\text{C} = \text{kg m}\text{s}^{-2}\text{A}^{-1}$.

Definition 1.3.15  A Telsa (T) is the SI unit of magnetic field defined by $\text{T} = \text{V s}/\text{m}^2 = \text{kg }\text{s}^{-2}\text{A}^{-1}$.

Definition 1.3.16  The fine structure constant denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value¹:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

Law 1.3.17  Maxwell's Equations are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]\[\nabla\cdot \vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}\]\[\nabla\times \vec{B} = \mu_0\left( \vec{J} + \varepsilon_0\frac{\partial \vec{E}}{\partial t} \right)\]

Definition 1.3.18  The vacuum permittivity $\varepsilon_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\varepsilon_0 = \frac{e^2}{2\alpha hc}\]

Definition 1.3.19  The vacuum permeability $\mu_0$ is the physical constant defined in terms of the fine structure constant $\alpha$, charge of an electron $e$, Plank constant $h$ and speed of light $c$.
\[\mu_0 = \frac{2\alpha h}{e^2c}\]

Result 1.3.20  The product of vacuum permittivity and vacuum permeability is the reciprocal of the speed of light squared.
\[\varepsilon_0\mu_0 = \frac{1}{c^2}\]

Definition 1.3.21  The Gaussian unit system is an alternate systems of units that uses the fundamental units of centimeters, grams, seconds and Kelvin to derive a system of units to describe the universe.

Definition 1.3.22  A second (s) is the SI unit of time that is exactly 9192631770 hyperfine transitions of a Caesium-133 atom.

Definition 1.3.23  A centimeter (cm) is the Gaussian unit of distance that is exactly the distance light travels in $1/29979245800$ seconds.

Definition 1.3.24  A gram (g) is the Gaussian unit of mass defined exactly by fixing Plank's constant $h=6.62607015 \times 10^{−27}\text{g } \text{cm}^2\text{s}^{−1}$.

Definition 1.3.25  A statcoulomb (statC) is the Gaussian unit of charge defined by $\text{statC} = \text{g}^{1/2}\text{cm}^{1/2}/\text{s}$ or exactly the amount of charge that results in one $\text{dyn}$ of repulsion between two particles of that charge separated by one $\text{cm}$.

Corollary 1.3.26  The vacuum permitivity in Gaussian units denoted $\varepsilon_0$ is unitless and $\varepsilon_0 = \frac{1}{4\pi}$.

Corollary 1.3.27  The vacuum permeability in Gaussian units denoted $\mu_0$ is unitless and $\mu_0=4\pi/c^2$.

Definition 1.3.28  The fine structure constant denoted $\alpha$ is a dimensionless experimentally determined constant defined below. In any system of units, the fine structure constant is dimensionless and therefore has the same value¹:
\[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\]

Result 1.3.29  Maxwell's equations in Gaussian Units are a set of coupled differential equations that form the foundations of classical electromagnetism.
\[\nabla\cdot\vec{E} = 4\pi \rho\]\[\nabla\cdot\vec{B} = 0\]\[\nabla\times\vec{E} = -\frac{1}{c} \frac{\partial \vec{B}}{\partial t}\]\[\nabla\times\vec{B} = \frac{1}{c}\frac{\partial \vec{E}}{\partial t} + \frac{4\pi}{c}\vec{J}\]

2 Electrostatics of ConductorsKW01

2.1 Microscopic Electric FieldWWKA

Result 2.1.1  The total energy of many conductors $U$ is the sum of the product of the charge on each conductor $q_i$ and there potentials $V_i$.
\[U = \frac{1}{2}\sum_iq_iV_i\]

Definition 2.1.2  The microscopic electric field denoted $\vec{e}(\vec{r})$ is the electric field with full spacial resolution down to the atomic details at position $\vec{r}$.

Definition 2.1.3  The spacial average denoted $\overline{f(\vec{r})}$ of a function $f(\vec{r})$ is defined for some characteristic distance $d$ by the following integral.$\newcommand\abs[1]{\left|#1\right|}$
\[\overline{f(\vec{r})} = \frac{1}{\sqrt{2\pi d}}\int f(\vec{r}')\exp\left(-\frac{\abs{\vec{r}-\vec{r}'}^2}{2d^2}\right) d^3r'\]

Definition 2.1.4  The macroscopic electric field denoted $\vec{E}(\vec{r})$ is the spatial average of the microscopic electric field $\vec{e}(\vec{r}')$.
\[\vec{E}(\vec{r}) = \overline{\vec{e}(\vec{r})}\]

Example 2.1.5  Macroscopic Electric Field in a 1D Crystal: Consider a periodic microscopic electric field in 1 dimension: $\vec{e}(x) = \sum_{k\in\mathbb{Z}}e^{2\pi i k x / a}$ for lattice spacing $a<<d$ much less than the averaging distance $d$.

Example 2.1.6  Calculation of the screening length in a conductor (Thomas–Fermi screening)

2.2 Mutual CapacitanceJ6R9

Result 2.2.1  The surface charge density of a conductor $\sigma$ can be written in terms of the change in direction on the electric field $\vec{E}$ in the outward facing normal $\vec{n}$ of the surface of a conductor.
\[\vec{E}\cdot\vec{n} = -\frac{\partial V}{\partial \vec{n}} = 4\pi\sigma(\vec{r})\]

Result 2.2.2  The total energy of many conductors $U$ is the sum of the product of the charge on each conductor $q_i$ and there potentials $V_i$.
\[U = \frac{1}{2}\sum_iq_iV_i\]

Definition 2.2.3  The mutual capacitance $C_{i,j}$ is a matrix describing the linear relationship between the potentials of many conductors $V_i$ and the total charges of many conductors $q_i$.
\[q_i = \sum_jC_{i,j}V_j\]

Result 2.2.4  The inverse of mutual capacitance can be used to find the potentials $V_i$ of many conductors from the charge on each conductor $q_i$.
\[V_i = \sum_jC^{-1}_{ij}q_j\]

Result 2.2.5  The total energy of many conductors at fixed potentials $U$ can be written in terms of the mutual capacitance $C_{i,j}$ and the potentials of each conductor $V_i$.
\[U = \frac{1}{2}\sum_{i}\sum_{j}V_iC_{i,j}V_j\]

Result 2.2.6  The following properties of mutual capacitance hold for any system of conductors.
\[C_{i,i}\geq 0,\quad C_{i,i}C_{j,j}\geq C_{i,j}^2,\quad C_{i\neq j}\leq 0\]

Example 2.2.7  Mutual Capacitance for a Sphere

Result 2.2.8  The following partial derivatives of energy of conductors fold for any system of conductors.
\[\frac{\partial U}{\partial q_i} = V_i,\quad \frac{\partial U}{\partial V_i} = q_i\]

2.3 Method of Images6TET

Definition 2.3.1  The method of images is a technique for solving the Laplace equation for a set of boundary conditions by places virtual charges outside the volume of interest.

2.4 Method of DisplacementsNHJ1

Definition 2.4.1  The method of displacements in Gaussian units can be used to calculate the force on a conductor by considering the change in energy caused by displacing an infinitesimal amount of electric field $\vec{E}$ produced by the surface charge density $\sigma$ on the surface of a conductor by moving an infinitesimal area $dA$ of the conductor. This can be integrated over the surface $S$ of a conductor to calcualte the total force.
\[d\vec{F} = \frac{E^2}{8\pi}d\vec{S} = 2\pi \sigma d\vec{S}\]\[F = \int_S \frac{E^2}{8\pi}d\vec{S} = \int_S 2\pi \sigma d\vec{S}\]

3 Electrostatics of DielectricsMA2Z

3.1 Polarization699A

Definition 3.1.1  The microscopic charge density denoted $\varrho(\vec{r})$ is the charge per unit volume with full spacial resolution down to the atomic details at position $\vec{r}$.

Definition 3.1.2  The macroscopic charge density denoted $\rho$ is the spacial average of the microscopic charge density $\varrho(\vec{r}')$.
\[\rho(\vec{r}) = \overline{\varrho(\vec{r})}\]

Definition 3.1.3  The bound charge density denoted $\rho_{b}$ is the density of charges that are part of a neutral material.
\[\rho_b = \rho - \rho_f\]

Definition 3.1.4  The free charge density or external charge density denoted $\rho_f$ or sometimes $\rho_{ex}$ is the density of charges that are not part of a neutral material.
\[\rho_f = \rho - \rho_b\]

Result 3.1.5  The integral over all space of the bound charge density is zero.
\[\iiint_V\rho_bdV = 0\]

Definition 3.1.6  The polarization denoted $\vec{P}$ is the electric dipole moment per unit volume of the bound charge density $\rho_b$ in a material.
\[\vec{P} = \frac{d\vec{p}}{dV}\]\[- \nabla\cdot\vec{P} = \rho_b\]

Definition 3.1.7  The electric displacement field denoted $\vec{D}$ is defined in Gaussian units in terms of the electric field $\vec{E}$ and the polarization $\vec{P}$.
\[\vec{D} = \vec{E} + 4\pi\vec{P}\]

Result 3.1.8  The divergence of the displacement field $\vec{D}$ in Gaussian units is the free charge density $\rho_f$.
\[\nabla\cdot\vec{D} =4\pi \rho_f\]

Law 3.1.9  Gauss's law for Displacement Field states that states that the surface integral of the electric displacement field $\vec{D}$ dotted with the normal vector $\vec{n}$ over a piecewise smooth boundary $S$ of a volume $V$ is the volume integral of the free charge density $\rho_f$ over a volume $V$.
\[\int_S (\vec{D}\cdot \vec{n}) d^2r = \int_V\rho_f(\vec{r})d^3r\]

Definition 3.1.10  The dipole moment $\vec{p}$ of a charge distribution $\rho(\vec{r})$ is defined with the following integral.
\[\vec{p} = \int\vec{r}\rho(\vec{r})d^3r\]

Result 3.1.11  The electric dipole moment $\vec{p}_f$ of a free charge density $\rho_f$ is the volume integral of the polarization $\vec{P}$.
\[\vec{p}_f = \int_V\vec{P}dV\]

3.2 Linear Dielectrics1K3W

Definition 3.2.1  A linear dielectric is a dielectric where the polarization $\vec{P}$ and electric displacement field $\vec{D}$ are linearly proportional to the electric field $\vec{E}$
\[\vec{P}\propto\vec{E},\quad \vec{D}\propto\vec{E}\]

Definition 3.2.2  An isotropic linear dielectric is a linear dielectric where the displacement field $\vec{D}$ is related to the electric field $\vec{E}$ by the permittivity scalar $\epsilon$.
\[\vec{D} = \varepsilon \vec{E}\]

Definition 3.2.3  An anisotropic linear dielectric is a linear dielectric where the displace field $\vec{D}$ is related to the elecctric field $\vec{E}$ by a non-trivial linear permittivity matrix $\mathcal{E}$.
\[ \vec{D} = \mathcal{E}\vec{E}\]

Definition 3.2.4  The polarizability $\kappa$ of an isotropic linear dielectric is defined in terms of the permittivity $\varepsilon$ by the following formula.
\[\kappa = \frac{\varepsilon - 1}{4\pi}\]

Result 3.2.5  The polarization $\vec{P}$ from a uniform electric field $\vec{E}$ in a linear dielectric is related by the polarizability $\kappa$.

\[\vec{P} = \kappa \vec{E}\]

Result 3.2.6  The surface charge density of a conductor in a dielectric $\sigma$ can be written in terms of the change in direction on the electric displacement field $\vec{D}$ in the outward facing normal $\vec{n}$ of the surface of a conductor.
\[\vec{D}\cdot\vec{n} = -\varepsilon\frac{\partial V}{\partial \vec{n}} = 4\pi\sigma(\vec{r})\]

Result 3.2.7  The boundary conditions of the surface of isotropic dielectrics are that the tangential components $\vec{E}_{\parallel 1},\vec{E}_{\parallel 2}$ of the electric field are continuous and the normal components $\vec{E}_{\perp 1},\vec{E}_{\perp 2}$ scales proportional to the permittivities $\varepsilon_1,\varepsilon_2$.
\[\vec{E}_{\parallel 1} = \vec{E}_{\parallel 2},\quad \varepsilon_1\vec{E}_{\perp 1} = \varepsilon_2 \vec{E}_{\perp 2}\]\[V_1 = V_2,\quad \varepsilon_1\frac{\partial V_1}{\partial n} = \varepsilon_2\frac{\partial V_2}{\partial n}\]

Example 3.2.8  Dielectric Image Charges: Consider an charge $q$ placed in an isotropic linear dielectric with permittivity $\varepsilon_1$ a distance $h$ away from an infinite plane interface of a second isotropic linear dielectrics with permittivity $\varepsilon_2$. The system can be solved with a different image charge for each region of dielectric. For the region occupied by the first dielectric $\varepsilon_1$, the image charge $q'$ is placed directly opposite the charge $q$ and the potential in this region is $V_1$.
\[q' = q\frac{\varepsilon_1- \varepsilon_2}{\varepsilon_1+\varepsilon_2}, \quad V_1 = \frac{q}{4\pi\varepsilon_1 r} + \frac{q'}{4\pi\varepsilon_1 r'}\]
For the region occupied by the second dielectric $\varepsilon_2$, the image charge $q''$ is placed directly on top of the original charge $q$ and the potential in this region is $V_2$.
\[q'' = q\frac{\varepsilon_2-\varepsilon_1}{\varepsilon_1+\varepsilon_2}, \quad V_2 = \frac{q + q''}{4\pi\varepsilon_2 r} = \frac{2q\varepsilon_2}{\varepsilon_1 + \varepsilon_2}\frac{1}{4\pi\varepsilon_2 r}\]

3.3 Thermodynamics of Dielectrics8MDM

3.3.1 

4 ConductivityNZ03

4.1 

5 MagnetostaticsW455

5.1 

6 AppendixFEZC

6.1 References6EZP

¹

Condensed Matter Physics

Quantum Transport and Mesoscopic Physics60ZR

1 IntroductionZ4MT

1.1  Recommended Textbooks

• J.M. Ziman Principles of the theory of solids, 2nd Edition
• A. Atland and B. Simons, Condensed Matter Field Theory
• S.M. Girvin and Kun Kang, Modern Condensed Matter Physics

1.2  Presentation Topics

• Coulomb blockade.
• Kondo effect
• Orthogonality catastrophe and quantum speed limit
• Mott transition
• Spin Hall effect
• Composite fermions and the fractional quantum Hall effect
• Charge density waves
• Strongly coupled large polarons

2 Brownian Motion0ZE2

2.1 Stationary 1D Brownian MotionDFZJ

Definition 2.1.1  A quantum particle exhibits coherent motion iff its lifetime $t_\ell$ is much greater than $\frac{\hbar}{E}$.
\[t_\ell >> \frac{\hbar}{E}\]

Definition 2.1.2  A quantum particle exhibits incoherent motion iff its lifetime $t_\ell$ less than $\frac{\hbar}{E}$.
\[t_\ell \leq \frac{\hbar}{E}\]

Definition 2.1.3  A stationary 1D Brownian particle is a system of a stationary 1D particle experiencing many fast collisions with momentum $p_i$ integrated over a small time step $\Delta t$. Defined the force experienced by the particle $f(t)$ as a function of time.
\[f(t) = \frac{2}{\Delta t}\sum_i{p_i}\]

Definition 2.1.4  The ensemble average denoted $\langle x \rangle$ of a random variable $x$ is the integral of the variable weighted by the probability density function $P_{x}(x)$ across all possible values.
\[\langle x \rangle = \int_{-\infty}^{\infty}{x P_{x} dx}\]

Definition 2.1.5  The momentum diffusion coefficient is defined $D=2\nu\langle p_i^2\rangle = 2\nu m_{mol} k_B T$ where $\nu$ is the average frequency of collisions and $m^*$ is the effective mass of the particles.

Result 2.1.6  $\langle f\rangle = 0$

Result 2.1.7  $\langle f^2\rangle = \frac{4}{(\Delta t)^2} \sum_i{\langle p_i^2 \rangle}=\frac{4}{\Delta t}\nu\langle p_i^2\rangle = \frac{2D}{\Delta t}$

Result 2.1.8  $\langle f^{2n+1} \rangle = 0$

Result 2.1.9  $\langle f^{2n} \rangle = \langle f^2 \rangle^n (2n-1)!! =\left(\frac{4}{\Delta t}\nu\langle p_i^2\rangle\right)^n (2n-1)!! = \left(\frac{2D}{\Delta t}\right)^n (2n-1)!!$

Definition 2.1.10  A Gaussian Distribution with variance $\sigma=\langle f^2 \rangle$ is a distribution of the form:
\[P_f(x) = \frac{1}{\sqrt{2\pi\sigma}}e^{-x^2/2\sigma}\]

Definition 2.1.11  A probability density function of a variable is a function that when integrated over a region of possible values represents the probability of the variable being in that region.

Result 2.1.12  The probability density function of $f(t)$ at a particular time is Gaussian with $\sigma = \langle f^2 \rangle = \frac{2D}{\Delta t}$.

Definition 2.1.13  A functional probability density function is a probability density function that is integrated over different functions.

Result 2.1.14  The functional probability density of $f(t)$ is \[\mathcal{P}(f(t)) = C \text{exp}\left[{-\frac{1}{4D}\int_{-\infty}^{\infty}{f^2(t)dt}}\right]\]

2.2 1D Brownian Motion with Friction9ET5

Definition 2.2.1  The friction coefficient is a constant that describes the amount of friction experienced by a 1D Brownian particle.
\[\Gamma = \frac{m_{mol}\nu}{M}\]

Definition 2.2.2  A 1D Brownian Particle is a system of a 1D particle of mass $m_p$ moving at velocity $v_{p}$ which is experiencing many fast collisions. Define the force experienced by the particle $f_p(t)$ in terms of the force experienced by a stationary 1D Brownian particle $f(t)$ and the friction coefficient $\Gamma$.
\[f_v(t) = f(t) - 2\Gamma m_p v_p\]

Definition 2.2.3  The position diffusion coefficient denoted $\mathscr{D}$ is defined in terms of the noise intensity $D$, the friction coefficient $\Gamma$ and the mass of the particle $m_p$.
\[\mathscr{D} = \frac{D}{(2\Gamma m_p)^2} = \frac{k_B T}{2\Gamma m_p}\]

Result 2.2.4  The motion $q(t)$ and the momentum $p(t)$ of a 1D Brownian particle is described by the following equations.
\[p(t) = \int_0^t{e^{-2\Gamma(t-\tau)f(\tau)d\tau}}\]\[q(t) = \frac{1}{m_p}\int_0^t{p(\tau)d\tau} = \frac{1}{m_p}\int_0^\tau{f(\tau)\frac{1-e^{-2\Gamma(t-\tau)}}{2\Gamma}d\tau}\]

Result 2.2.5  $\langle q\rangle = 0$

Result 2.2.6  For $t>>\frac{1}{\Gamma}$, $\langle q^2\rangle = \langle (q(t)-q(0))^2 \rangle = 2\mathscr{D}\left[t - \frac{2}{2\Gamma}(1-e^{-2\Gamma t}) + \frac{1}{4\Gamma}(1-e^{-4\Gamma t})\right] \approx 2\mathscr{D}t$

2.3 Lagevin and Fokker-Plank Equations26RC

Definition 2.3.1  A 1D Brownian Particle in a Potential is a system of a 1D Brownian particle experiencing a potential $U(p)$. The system can be written in terms of position $q$ and momentum $p$ or as a time dependent force function $f_U(t)$
\[\dot{q} = \frac{p}{m_p},\quad \dot{p} = f(t) - 2\Gamma p - U'(q)\]\[f_U(t) = f(t) - 2\Gamma m_pv_p - U'\]

Theorem 2.3.2  The Lagevin and Fokker-Plank Equations describe the probability density function $\omega(q,p,t,q(0),p(0),0)$ of a 1D Brownian particle in a potential $U(q)$.
\[\omega(q,p,0,q(0),p(0),0) = \delta(q-q(0))\delta(p-p(0))\]\[\partial_t \omega(q,p,t)= -\partial_q\left( \frac{p}{m_p}\omega(q,p,t) \right) - \partial_p\left(- 2\Gamma p - U'(q) + D\partial_p^2\right)\omega(q,p,t)\]

Definition 2.3.3  The Diffusion Equation describes a Brownian particle where there is no potential or drift, that is $\Gamma = 0$ and $U(q)=0$.
\[\frac{\partial n}{\partial t} - \mathscr{D}\nabla_\mathbf{x}^2n = 0\]

Result 2.3.4  The solution to the diffusion equation is of the form
\[\partial_t\omega = \mathscr{D}\partial_x^2\omega\]\[\omega = \frac{C}{\sqrt{t}}e^{-x^2/4\mathscr{D}t}\]

Definition 2.3.5  The time average denoted $\langle x \rangle$ of a random variable $x(t)$ is the time integral of the variable over a very long period.
\[\langle x\rangle = \lim_{T\to\infty}\frac{1}{T}\int_0^T{x(t)dt}\]

Definition 2.3.6  A system is ergodic iff the ensemble average and the time average are equivalent.

2.4 Harmonic Oscillator DiffusionK69N

Definition 2.4.1  The Brownian harmonic oscillator is a system describing a Brownian particle in a harmonic oscillator potential $U(q)$
\[U(q) = \frac{1}{2}m\omega_0^2q^2\]\[\partial_t \omega(q,p,t) = -\partial_q\left( \frac{p}{m_p}\omega(q,p,t) \right) - \partial_p\left(- 2\Gamma p\omega(q,p,t) - m\omega_0^2q\omega(q,p,t) + D\partial_p^2\right)\omega(q,p,t)\]

Result 2.4.2  For a Brownian harmonic oscillator, $\langle \ddot{q}\rangle = -\omega_0^2\langle q \rangle - \frac{2\Gamma}{m}\langle \ddot{q}\rangle$

Definition 2.4.3  The time correlation function $Q_{qq}(t)$ of a variable $q(t)$ is defined $Q_{qq}(t) = \langle q(t)q(0)\rangle$.

Definition 2.4.4  The power spectrium $S_{qq}(\omega)$ of a variable $q(t)$ is the Fourier transform of the time correlation function $S_{qq}(\omega) = \int_{\infty}^\infty{e^{i\omega t}\langle q(t) q(0)\rangle dt}$,

Result 2.4.5  For a real variable $q(t)$ the power spectrum is $S_{qq}(\omega) = 2\text{Re}\left( \int_{-\infty}^\infty{e^{i\omega t}\langle q(t) q(0)\rangle dt} \right)$

Result 2.4.6  The stationary solution to the Brownian Harmonic oscillator is of the form
\[\omega(q,p,t) = C e^{-(\frac{p^2}{2m} + U(q))/\alpha}\]

Result 2.4.7  The power spectrum of $q$ for the Brownian harmonic oscillator is
\[S_qq(\omega) = \frac{2\Gamma \omega_0^2}{(\omega^2-\omega_0^2)^2 + 4\Gamma^2\omega^2}\]

2.5 Escape and Activation via Tunneling8553

Definition 2.5.1  The probability current $j(q)$ is the rate at which probability to crossing point $q_B$.
\[j(q_B) = \int{\frac{p}{m}\omega(q_B,p,t)}\]

Definition 2.5.2  The escape rate is the reciprocal of the average lifetime $\tau$ and can be written in terms of probability current.
\[W_{esc} = \frac{j}{N} = \frac{1}{\tau}\]

Definition 2.5.3  The number of trapped particles $N$ is defined
\[N = \int_{-\infty}^{\infty}dp\int_{-\infty}^{q_B}dq \omega(q,p,t)\]

Result 2.5.4  The time derivative of the number of particles $N$ is $\frac{\partial N}{\partial t} = - W_{esc}N$ and the decay over time is
\[N(t) = N(0)e^{-W_{esc}t}\]

2.6 Escape and Activation via Tunneling8553

Definition 2.6.1  The probability current $j(q)$ is the rate at which probability to crossing point $q_B$.
\[j(q_B) = \int{\frac{p}{m}\omega(q_B,p,t)}\]

Definition 2.6.2  The escape rate is the reciprocal of the average lifetime $\tau$ and can be written in terms of probability current.
\[W_{esc} = \frac{j}{N} = \frac{1}{\tau}\]

Definition 2.6.3  The number of trapped particles $N$ is defined
\[N = \int_{-\infty}^{\infty}dp\int_{-\infty}^{q_B}dq \omega(q,p,t)\]

Result 2.6.4  The time derivative of the number of particles $N$ is $\frac{\partial N}{\partial t} = - W_{esc}N$ and the decay over time is
\[N(t) = N(0)e^{-W_{esc}t}\]

3 Electron TransportZKM7

3.1 Bloch's Theorem and Berry Phase45C9

Theorem 3.1.1  Bloch's Theorem states that the solution any system with a periodic potential can be represented with a periodic function $u_{\mathbf{k},n}(\mathbf{r})$ and $e^{i\mathbf{k}\cdot\mathbf{r}}$, that is
\[H\Psi_{\mathbf{k},n}(\mathbf{r}) = E_{\mathbf{k},n}\Psi_{\mathbf{k},n}(\mathbf{r}),\quad \Psi_{\mathbf{k},n}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k},n}(\mathbf{r})\]

Definition 3.1.2  The Berry connection denoted $\mathcal{A}_{n,\mathbf{k}}(\mathbf{r})$ is defined as $\mathcal{A}_{n,\mathbf{k}}(\mathbf{r}) = i\langle u_{\mathbf{k},n}(\mathbf{r}) | \nabla_{\mathbf{k}} u_{\mathbf{k},n}(\mathbf{r}) \rangle$

Definition 3.1.3  The Berry phase denoted $\gamma_{n,\mathbf{k}}$ is defined as $\gamma_{n,\mathbf{k}} = \int \mathcal{A}_{n,\mathbf{k}}(\mathbf{r}) \cdot d\mathbf{r}$

Result 3.1.4  For the band wavefunction $\Phi_n(\mathbf{r}) = \int{ c(\mathbf{k})\Psi_{\mathbf{k},n}(\mathbf{k})\Psi_{\mathbf{k},n}(\mathbf{r}) d\mathbf{k}}$, written in terms of eigenstates $\Psi_{\mathbf{k},n}$,
\[\mathbf{r} = i\partial_k + i\langle u_{\mathbf{k},n}(\mathbf{r}) | \nabla_{\mathbf{k}} u_{\mathbf{k},n}(\mathbf{r}) \rangle = i\partial_k + \mathcal{A}_n(\mathbf{r})\]\[\langle \mathbf{r} \rangle = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k}) \mathbf{r} c(\mathbf{k}) d\mathbf{k}} = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k})\left(i\partial_k + \mathcal{A}_{n,\mathbf{k}}(\mathbf{r})\right) c(\mathbf{k}) d\mathbf{k}}\]\[\langle \dot{\mathbf{r}} \rangle = \frac{\partial E_{\mathbf{k},n}}{\partial \mathbf{k}}\]

3.2 Anomalous Quantum Hall Effect63P4

Definition 3.2.1  The anomalous quantum hall effect Hamiltonian $H$ describes the behavior of many electrons in a lattice $H_0$ experiences an electric field $E$ pointing in the x direction.
\[H = H_0 - e_qEx\]

Definition 3.2.2  The Berry curvature denoted $\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r})$ is defined as $\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r}) = \nabla_\mathbf{r}\times \mathcal{A}_{n,\mathbf{k}}(\mathbf{r})$

Result 3.2.3  The average velocities for 2D electrons experiencing an electric field in the x direction are
\[\langle v_x\rangle = \frac{1}{\hbar} \frac{\partial E_{n\mathbf{k}}}{\partial k_x}\]\[\langle v_y \rangle = \frac{1}{\hbar} \frac{\partial E_{n\mathbf{k}}}{\partial k_x} + \frac{qE}{\hbar}\left(\mathbf{\Omega}_{n,\mathbf{k}}(\mathbf{r})\right)_z\]

Result 3.2.4  current

Definition 3.2.5  Churn number

Result 3.2.6  conductivity

3.3  Boltzmann Kinetic Equation

4 Scattering and ConductivityREZC

4.1  Impurity Scattering

4.2  Electric Conductivity

4.3  Thermal Conductivity

4.4  Magnetoconductivity

4.5  Cyclotron Resonance

5 Low-dimensional SystemsAW5M

5.1  Landauer 1D Conductivity

5.2  2D Electron Systems

5.3  Quantum Hall Effect

6 LocalizationD3P7

6.1  Weak Localization

6.2  Anderson Localization

6.3  Density Matrix and the Quantum Kinetic Equation

7 Phonon1EDE

7.1  Electron-phonon Interaction

7.2  Polaronic Effect

7.3  Ohmic Dissipation

7.4  The Orthogonality Catastrophe

7.5  Holstein Polarons

8 Topological MaterialsNMJH

8.1  Variable-range Hopping

8.2  The Coulomb Gap

8.3  The Berry Phase

8.4  Group Velocity in Topologically Nontrivial Solids

8.5  The Kitaev Chain

Math64CC

Numerical Linear Algebra2CZ4

1 Fundamental Linear AlgebraZ6PT

1.1 Vector SpacesD28A

Definition 1.1.1  A Vector Space is a set $V$ equipped with addition and scalar multiplication with the following properties

1. Commutativity: $\mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v}$, $\forall\ \mathbf{v},\mathbf{w}\in V$
2. Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})$, $\forall\ \mathbf{u},\mathbf{v},\mathbf{w}\in V$
3. Zero Vector: $\exists\ \mathbf{0}\in V$ such that $\mathbf{v}+\mathbf{0}=\mathbf{v}$, $\forall\ \mathbf{v}\in V$
4. Additive inverse: For every $\mathbf{v}\in V$, there exists $-\mathbf{v}\in V$ such that $\mathbf{v}-\mathbf{v}=\mathbf{0}$
5. Multiplicative Identity: $1\mathbf{v} = \mathbf{v}$ for all $\mathbf{v}\in V$
6. Multiplicative Associativity: $(\alpha\beta)\mathbf{v}$ = $\alpha(\beta\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha, \beta$
7. Additive Closure: $\mathbf{v}+\mathbf{w}\in V$ for all $\mathbf{v},\mathbf{w}\in V$
8. Multiplicative Closure: $\alpha\mathbf{v}\in V$ for all $\mathbf{v}\in V$ and scalars $\alpha$
9. $\alpha(\mathbf{u}+\mathbf{v}) = \alpha\mathbf{u} + \alpha\mathbf{v}$ for all $\mathbf{u},\mathbf{v}\in V$ and scalars $\alpha$
10. $(\alpha + \beta)\mathbf{v}=\alpha\mathbf{v}+\beta\mathbf{v}$ for all $\mathbf{v}\in V$ and scalars $\alpha, \beta$

Proposition 1.1.2  $\mathbb{C}^n$ is a vector space with scalars in $\mathbb{C}$.

Definition 1.1.3  A subspace of a vector space $V$ is a subset $S\subset V$ that has additive closure and multiplicative closure.

Definition 1.1.4  A linear combination of a subset of vectors $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_n\}\subset V$ for scalars $\{a_1,a_2,\dots,a_n\}$ is the sum
\[a_1\mathbf{v}_1+a_2\mathbf{v}_2 + \dots + a_n\mathbf{v}_n = \sum_{i=1}^na_i\mathbf{v}_i\]

Definition 1.1.5  A linearly independent set of vectors $\{\mathbf{v}_i\}$ is a set of vectors such that only trivial linear combinations produce the zero vector $\mathbf{0}$.

Definition 1.1.6  A linearly dependent set of vectors $\{\mathbf{v}_i\}$ is a set of vectors such that there exists a non-trivial linear combination.

Definition 1.1.7  The span of a set of vectors is the set of all possible linear combinations.

Definition 1.1.8  A basis of a vector space $V$ is a set of vectors with span $V$ that is linearly independent.

Definition 1.1.9  The dimension denoted $\text{Dim}(V)$ of a vector space $V$ is the number of vectors in any basis of $V$.

Definition 1.1.10  The inner product denoted $\mathbf{v}\cdot\mathbf{w}\in\mathbb{C}$ of two vectors $\mathbf{v},\mathbf{w}\in V$ is the sum of the product of there elements.
\[\mathbf{v}\cdot\mathbf{w} = \langle \mathbf{v},\mathbf{w} \rangle = \mathbf{v}^*\mathbf{w} = \sum_i{\bar{v_i}w_i} \in \mathbb{C}\]

Definition 1.1.11  The outer product denoted $\mathbf{v}\mathbf{w}^*\in\mathbb{C}^{n\times n}$ of two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}$ is the matrix of products of there elements.
\[\mathbf{v}\mathbf{w}^* = (v_iw_j) \in \mathbb{C}^{n\times n}\]

Definition 1.1.12  The elementary vectors denoted $\mathbf{e}_i\in\mathbb{C}^n$ are the vectors with $e_i = 1$ and zeros for all other elements.

1.2 MatricesZ05T

Definition 1.2.1  A linear transformation is a function $T:V\to W$ for vector fields $V,W$ such that

1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v})+ T(\mathbf{w})$ for all $\mathbf{v},\mathbf{w}\in V$
2. $T(\alpha\mathbf{v}) = \alpha T(\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha$

Definition 1.2.2  A matrix is an grid of entries denoted as a rectangular array, a series of column vectors or a series of row vectors.

Definition 1.2.3  Matrix-vector multiplication denoted $M\mathbf{v}\in V_m$ for a matrix $M=(\mathbf{t}_1,\mathbf{t}_2,\dots,\mathbf{t}_n)\in M_{m\times n}$ and a vector $\mathbf{v}\in V_n$ is
\[M\mathbf{v} = \sum_{i=1}^{n}{v_i \mathbf{m}_i}\quad\text{where } \mathbf{m}_i \text{ are the column vectors of }T\]

Definition 1.2.4  Matrix multiplication denoted $AB\in M_{m\times n}$ for two matrices $A \in M_{m\times k}$ and $B\in M_{k\times n}$ is the matrix where the elements are the dot products of the row vectors of $A$ and the column vectors of $B$. Let $\mathbf{a}_i$ be the row vectors of $A$ and $\mathbf{b}_j$ be the column vectors of $B$, then the elements of $AB$ matrix are
\[AB_{i,j} = (\mathbf{a}_i\cdot\mathbf{b}_j)\]

Proposition 1.2.5  Any matrix $M\in \mathbb{C}^{m\times n}$ defines a linear transformation $M:\mathbb{C}^n\to\mathbb{C}^m$ defined by matrix vector multiplication.

Proposition 1.2.6  Matrix multiplication is the same as function composition of the corresponding linear transformations.

Definition 1.2.7  The conjugate denoted $\overline{M}$ of a matrix $M=(m_{i,j})\in\mathbb{C}^{m\times n}$ is the matrix with complex conjugate elements of $M$.
\[\overline{M} = (\bar{m}_{i,j})\]

Definition 1.2.8  The transpose denoted $M^T$ of a matrix $M=(m_{i,j})\in\mathbb{C}^{m\times n}$ is the matrix with swapped rows and columns of $M$.
\[M^T = (m_{j,i})\]

Definition 1.2.9  The adjoint denoted $M^*$ of a matrix $M=(m_{ij})\in\mathbb{C}^{n\times m}$ is conjugate transpose of $M$.
\[M^* = \overline{M}^T = (\bar{m}_{ji})\]

Definition 1.2.10  The inner product denoted $\mathbf{v}\cdot\mathbf{w}\in\mathbb{C}$ of two vectors $\mathbf{v},\mathbf{w}\in V$ is the sum of the product of there elements.
\[\mathbf{v}\cdot\mathbf{w} = \langle \mathbf{v},\mathbf{w} \rangle = \mathbf{v}^*\mathbf{w} = \sum_i{\bar{v_i}w_i} \in \mathbb{C}\]

Definition 1.2.11  The outer product denoted $\mathbf{v}\mathbf{w}^*\in\mathbb{C}^{n\times n}$ of two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}$ is the matrix of products of there elements.
\[\mathbf{v}\mathbf{w}^* = (v_iw_j) \in \mathbb{C}^{n\times n}\]

Definition 1.2.12  The elementary vectors denoted $\mathbf{e}_i\in\mathbb{C}^n$ are the vectors with $e_i = 1$ and zeros for all other elements.

Definition 1.2.13  The identity matrix is the matrix $I\in\mathbb{C}^{n\times n}$ such that $\mathbf{v} = I\mathbf{v}$ for all $\mathbf{v}\in \mathbb{C}^n$.
\[I = (\mathbf{e}_1,\mathbf{e}_2,\dots,\mathbf{e}_n)\]

Definition 1.2.14  The Kronecker delta $\delta_{ij} = \left\{\begin{array}{lr}
1, & \text{if } i = j\\
0, & \text{if } i\neq j
\end{array}\right\}$.

Definition 1.2.15  The kernel or null space denoted $\text{Ker}(M)$ of a matrix $M$ is the set of vectors $\{ \mathbf{v}\in V : M\mathbf{v} = \mathbf{0} \}$ that map to zero.

Definition 1.2.16  The nullity denoted $\text{Nullity}(M)$ of a matrix $M$ is the dimension of the kernel.

Definition 1.2.17  The range denoted $\text{Ran}(M)$ of a matrix $M$ is the set of vectors $\{ \mathbf{w}\in W : \exists \mathbf{v}\in V\text{ st. }M\mathbf{v}=\mathbf{w} \}$ in the span of the columns of $M$.

Definition 1.2.18  The rank denoted $\text{Rank}(M)$ of a matrix $M$ is the dimension of the range.

Proposition 1.2.19  The kernel and range of a matrix are subspaces of there respective vector spaces.

Theorem 1.2.20  The rank nullity theorem states that the sum of the rank and the nullity is equal to the number of columns.
\[\text{Rank}(M)+\text{Nullity}(M) = \text{ # of columns of M}\]

Definition 1.2.21  A matrix $M$ is invertible iff there exists a matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$.

Definition 1.2.22  The determinant denoted $\text{det}(A)\in\mathbb{C}$ of a matrix $A\in\mathbb{C}^{n\times n}$ with elements $a_{ij}\in\mathbb{C}$ is
\[\text{det}(A) = \sum_{\sigma\in\text{Perm}(\{1,2,\dots, n\})} a_{\sigma(1),1}a_{\sigma(2),2},\dots a_{\sigma(n),n}(-1)^{K(\sigma)}\]

Theorem 1.2.23  For any $A\in\mathbb{C}^{n\times n}$, the following are equivalent:

1. $A$ is invertible.
2. $\text{Rank}(A) = n$
3. $\text{Range}(A) = \mathbb{C}^n$
4. $\text{Ker}(A) = \emptyset$
5. $\text{det}(A)\neq 0$

Definition 1.3  A linear transformation is a function $T:V\to W$ for vector fields $V,W$ such that

1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v})+ T(\mathbf{w})$ for all $\mathbf{v},\mathbf{w}\in V$
2. $T(\alpha\mathbf{v}) = \alpha T(\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha$

1.4 Orthogonal and Orthonormal VectorsP8NF

Definition 1.4.1  The 2-norm denoted $||\mathbf{v}||$ of a vector $\mathbf{v}\in\mathbb{C}^n$ is the real number $||\mathbf{v}|| = \sqrt{\langle \mathbf{v}, \mathbf{v}\rangle}$

Definition 1.4.2  Two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ are orthogonal iff $\langle \mathbf{v},\mathbf{w}\rangle = 0$.

Definition 1.4.3  Two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ are orthonormal iff they are orthogonal and $||\mathbf{v}||=||\mathbf{w}||=1$.

Definition 1.4.4  The distance between two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ is the 2-norm of their difference $||\mathbf{v}-\mathbf{w}||$.

Definition 1.4.5  The angle between two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ is $\arccos\frac{\langle \mathbf{v},\mathbf{w} \rangle}{||\mathbf{v}||\cdot||\mathbf{w}||}$.

Definition 1.4.6  The orthogonal subspace denoted $W^{\perp}\subset \mathbb{C}^n$ of a subset $W\subset\mathbb{C}^n$ is the subspace of vectors that are orthogonal to $W$.

Proposition 1.4.7  Orthogonal vectors are linearly independent.

Proposition 1.4.8  If $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v})n\}\subset\mathbb{C}^n$ is orthonormal then any vector $\mathbf{v}\in\mathbb{C}^n$ can be written as $\mathbf{v} = \langle \mathbf{v}_1, \mathbf{v}\rangle\mathbf{v}_1 + \langle \mathbf{v}_2, \mathbf{v}\rangle\mathbf{v}_2 + \dots + \langle \mathbf{v}_n, \mathbf{v}\rangle\mathbf{v}_n$.

Proposition 1.4.9  If $\{\mathbf{v}_1,\mathbf{v}_2,\dots\}$ is a basis of a subspace $W\subset \mathbb{C}^n$ and $\{\mathbf{w}_1,\mathbf{w}_2,\dots\}$ is a basis of $W^\perp\subset \mathbb{C}^n$, then $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{w}_1,\mathbf{w}_2,\dots\}$ is a basis of a $\mathbb{C}^n$ and $\dim(W)+\dim(W^{\perp}) = n$.

Proposition 1.4.10  For any matrix $A\in\mathbb{C}^{m\times n}$,

1. $\text{Ker}(A)^\perp = \text{Ran}(A^*)$
2. $\text{Rang}(A)^\perp = \text{Ken}(A^*)$

1.5 Hermitian and Unitary Matrices1JJJ

Definition 1.5.1  A matrix $A\in\mathbb{C}^{n\times n}$ is Hermitian iff $A^*=A$.

Definition 1.5.2  A matrix $A\in\mathbb{C}^{n\times n}$ is unitary iff $A^*A=AA^*=I$.

Proposition 1.5.3  If $Q\in\mathbb{C}^{n\times n}$ is unitary, then

1. $||Q\mathbf{u}||=||\mathbf{u}||, \quad\forall \mathbf{u}\in\mathbb{C}^n$.
2. The rows and columns of $Q$ are an orthonormal basis of $\mathbb{C}^n$.

Theorem 1.5.4  Eigenvalue Decomposition of Hermitian Matrices Theorem states that any Hermitian matrix $A\in\mathbb{C}^{n\times n}$ can be diagonalized by an Hermitian matrix $Q\in\mathbb{C}^{n\times n}$ that is
\[A = Q\begin{pmatrix}\lambda_1 && \ &&\ \\ \ && \ddots&&\ \\\ &&\ &&\lambda_n\end{pmatrix}Q^*\]
where $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $A$ and the columns of $Q$ are eigenvectors of $A$.

1.6 Singular Value DecompositionM6C4

Definition 1.6.1  A singular value of a matrix $A\in\mathbb{C}^{n\times m}$ is a real number $\sigma > 0$ such that $\sigma^2$ is an eigenvalue of $A^*A$ or $AA^*$.

Theorem 1.6.2  Full Singular Value Decomposition Theorem states that for any $A\in\mathbb{C}^{n\times m}$, there exists unitary $U\in\mathbb{C}^{n\times n}$, unitary $V\in\mathbb{C}^{m\times m}$ and semi-diagonal $\Sigma\in\mathbb{C}^{n\times m}$ such that
\[A = U\Sigma V^*\]
where the first $r$ diagonal elements of $\Sigma$ are the singular values $\sigma_1,\dots,\sigma_r$ of $A$ and the remaining elements of $\Sigma$ are zero.

Theorem 1.6.3  Reduced Singular Value Decomposition Theorem states that for any $A\in\mathbb{C}^{n\times m}$, there exists $U\in\mathbb{C}^{n\times r}$, unitary $V\in\mathbb{C}^{r\times m}$ and diagonal $\Sigma\in\mathbb{C}^{r\times r}$ such that
\[A = U\Sigma V^*\]
where the diagonal elements of $\Sigma$ are the singular values $\sigma_1,\dots,\sigma_r$ of $A$.

Proposition 1.6.4  For any matrix $A\in\mathbb{C}^{n\times m}$, the eigenvalue decomposition of $A^*A = U(\Sigma^*\Sigma)U^*$ and the eigenvalue decomposition $AA^*=V(\Sigma\Sigma^*)V^*$ determine the singular value decomposition $A=U\Sigma V^*$.

Proposition 1.6.5  The number of singular values of a matrix is the rank.

Proposition 1.6.6  For any matrix $A\in\mathbb{C}^{n\times m}$ with full singular value decomposition $A=U\Sigma V^*$ and any vector $x\in\mathbb{C}^m$
\[Ax = \sum_{j=1}^r{\sigma_j\langle v_j,x\rangle}u_j\]
where $\sigma_1,\dots,\sigma_r$ are the singular values of $A$, $v_j$ are the columns of $V$ and $u_j$ are the columns of $U$.

1.7 NormsHNPM

Definition 1.7.1  A norm is a function $||\cdot||:V\to[0,\infty)$ with the following properties:

1. Positive definiteness: $||v||\geq0$, $\forall v\in V$
2. Scaling: $||cv||=|c||||v|$, $\forall v\in V$ and scalars $c$
3. Trianglular inequality: $||u+v||\leq ||u||+||v||$, $\forall u,v\in V$

Definition 1.7.2  The p-norm denoted $||\mathbf{v}||_p$ of a vector $\mathbf{v}\in\mathbb{C}^n$ for $1\leq p <\infty$ is the real number $||\mathbf{v}||_p = \left(\sum_{i=1}^n{|v_i^p|}\right)^{\frac{1}{p}}$.

Definition 1.7.3  The $\infty$-norm denoted $||\mathbf{v}||_\infty$ of a vector $\mathbf{v}\in\mathbb{C}^n$ is the real number $||\mathbf{v}||_\infty = \max_{1\leq j \leq n}|v_j|$.

1.7.4 

Proposition 1.7.5  Holder's Inequality states that for $1\leq p, q < \infty$ such that $\frac{1}{p}+\frac{1}{q}=1$, then for any $x,y\in\mathbb{C}^n$,
\[|x\cdot y|\leq||x||_p||y||_q\]

1.8 Matrix Norms1AHP

Definition 1.8.1  The matrix norm denoted $||A||$ of a matrix $A\in\mathbb{C}^{m\times n}$ induced by a vector norm $||\cdot||$ is the real number $||A|| = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||}{||x||}=\max_{||x||=1}||Ax||$

Definition 1.8.2  The matrix p-q-norm denoted $||A||_{p,q}$ of a matrix $A\in\mathbb{C}^{m\times n}$ for $1\leq p,q \leq \infty$ is the real number $||A||_{p,q} = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||_p}{||x||_q}=\max_{||x||_q=1}||Ax||_p$

Proposition 1.8.3  For two matrices $A\in\mathbb{C}^{m\times k}$ and $B\in\mathbb{C}^{k\times n}$, the following inequality holds for any $1\leq p,q,r\leq \infty$.
\[||AB||_{p,r}\leq ||A||_{p,q}||B||_{q,r}\]

Proposition 1.8.4  The matrix $1$-norm is the max of the column sums. For any matrix $A\in\mathbb{C}^{m\times n}$ with column vectors $\{\mathbf{c}_1,\dots,\mathbf{c}_n\}$, $||A||_1 = \max_{j\in \{1,\dots,n\}}||\mathbf{c}_j||_1$.

Proposition 1.8.5  Matrix multiplication by unitary matrices preserves $2$-norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_2 = ||AV||_2 = ||A||_2\]

Definition 1.8.6  The Frobenius norm denoted $||A||_F$ of a matrix $A\in\mathbb{C}^{m\times n}$ is the real number $||A||_F = \sqrt{\text{Tr}(A^*A)} = \sqrt{\sum_{i=1}^m{\sum_{j=1}^n{|A_{i,j}|^2}}}$.

Proposition 1.8.7  Matrix multiplication by unitary matrices preserves Frobenius norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_F = ||AV||_F = ||A||_F\]

Corollary 1.8.8  For $A\in\mathbb{C}^{m\times n}$ with singular values $\sigma_1,\dots,\sigma_r$, $||A||_F = \sqrt{\sigma_1^2+\dots+\sigma_r^2}$

1.8.9 

1.9 Orthogonal ProjectorsFZ2N

Definition 1.9.1  The projection denoted $\text{Proj}_W(\mathbf{v})\in W$ of a vector $\mathbf{v}\in\mathbb{C}^n$ onto a subspace $W\subset \mathbb{C}^n$ is defined for an orthonormal basis $\{\mathbf{u}_1,\dots\mathbf{u}_r\}$ of $W$ by
\[\text{Proj}_W(\mathbf{v}) = \langle \mathbf{u}_1,\mathbf{v} \rangle \mathbf{v} + \dots + \langle \mathbf{u}_r,\mathbf{v} \rangle \mathbf{v} \]

Proposition 1.9.2  The projections $\text{Proj}_{W}(v),\text{Proj}_{W^\perp}(v)$ of $\mathbf{v}\in \mathbb{C}^n$ onto $W,W^\perp\subset\mathbb{C}^n$ are orthogonal.

Definition 1.9.3  An orthogonal projector matrix is a matrix $P\in\mathbb{C}^{n\times n}$ such that $P^2 = P^* = P$.

Proposition 1.9.4  If $P$ is an orthogonal projector, then $I-P$ is also an orthogonal projector.

Proposition 1.9.5  The product $\mathbf{u}\mathbf{u}^*$ for $\mathbf{u}\in\mathbb{C}^n$ such that $||\mathbf{u}||_2=1$ is an orthogonal projector that projects onto the span of $\mathbf{u}$.

Proposition 1.9.6  If $W\subset \mathbb{C}^n$ is a subspace with an orthonormal basis $\{\mathbf{u}_1,\dots,\mathbf{u}_k\}$, then the matrix $P=\mathbf{u}_1\mathbf{u}_1^*+\dots+\mathbf{u}_k\mathbf{u}_k^*$ is the orthogonal projector $\text{Proj}_W$ and $\text{Ran}(P)=W$.

Proposition 1.9.7  If $P\mathbf{v}=\text{Proj}_W(\mathbf{v})\ \forall \mathbf{v}\in\mathbb{C}^n$, then $(I-P)\mathbf{v} = \text{Proj}_{W^\perp}(\mathbf{v})$.

Proposition 1.9.8  If $P\mathbf{v}=\text{Proj}_W(\mathbf{v})\ \forall \mathbf{v}\in\mathbb{C}^n$, then $(I-P)\mathbf{v} = \text{Proj}_{W^\perp}(\mathbf{v})$.

2 Matrix DecompositionKHZ6

2.1 QR DecompositionT2RK

Definition 2.1.1  The full QR decomposition of a matrix $A\in\mathbb{C}^{m\times n}$ is the unitary matrix $Q\in\mathbb{C}^{m\times m}$ and the upper triangular matrix $R\in\mathbb{C}^{m\times n}$ such that
\[A = QR\]

Definition 2.1.2  The reduced QR decomposition of a matrix $A\in\mathbb{C}^{m\times n}$ is the unitary matrix $Q\in\mathbb{C}^{m\times n}$ and the upper triangular matrix $R\in\mathbb{C}^{n\times n}$ such that
\[A = QR\]

Theorem 2.1.3  For any $A\in\mathbb{C}^{m\times n}$ with $\text{Rank}(A)=n$, there exists a unique QR decomposition with $r_{i,i}>0$ for $i\in\{1,\dots,n\}$. If $\{a_1,\dots,a_n\}$ are the linearly independent columns of $A$, $\{q_1,\dots,q_n\}$ is an orthonormal basis such that $\text{span}\{q_a,\dots,q_k\} = \text{span}\{a_1,\dots,a_k\}$, for all $k\in\{1,\dots,n\}$. Then the matrix $R$ with elements,
\[r_{i,j} = \begin{cases}\langle q_i, a_j \rangle & i \leq j \\ 0 & i>j\end{cases}\]
and the matrix $Q$ with columns $\{q_1,\dots,q_n\}$ is the unique QR decomposition of $A$.

Proposition 2.1.4  For any orthonormal set of vectors $\{q_1,\dots,q_n\}$
\[\text{Proj}_{\text{span}^\perp\{q_1,\dots,q_n\}} = \text{Proj}_{\text{span}^\perp\{q_1\}}(\text{Proj}_{\text{span}^\perp\{q_2\}}((\dots))\]

Algorithm 2.1.5  The classical Gram-Schmidt algorithm can be used to calculate the $QR$ decomposition of a matrix $A$.

Algorithm 2.1.6  The modified Gram-Schmidt algorithm can be used to calculate an orthonormal basis $\{q_1,\dots,q_n\}$ from a set of vectors $\{a_1,\dots,a_n\}$ as well as the not normalized basis $\{v_1,\dots,v_n\}$. This algorithm is slightly modified to avoid numerical errors.

Proposition 2.1.7  For any orthonormal set of vectors $\{q_1,\dots,q_n\}$
\[\text{Proj}_{\text{span}^\perp\{q_1,\dots,q_n\}} = \text{Proj}_{\text{span}^\perp\{q_1\}}(\text{Proj}_{\text{span}^\perp\{q_2\}}((\dots))\]

2.2 Householder QR Decomposition7PZ8

Definition 2.2.1  The Householder QR decomposition of a matrix $A\in\mathbb{C}^{m\times n}$ is a set of unitary matrices $\{Q_n,\dots,Q_1\}\subset\mathbb{C}^{m\times m}$ such that the matrix $R\in\mathbb{C}^{m\times n}$ defined below is upper triangular.
\[R = Q_n\dots Q_1A\]

Definition 2.2.2  The Householder reflector $H_v\in\mathbb{C}^{n\times n}$ for a unit vector $v\in\mathbb{C}^n$ is the following matrix.
\[H_v = I-2vv^*\]

Proposition 2.2.3  For $x,y\in\mathbb{R}^n$ with $||x||=||y||$, and $x\neq y$, the Householder reflector $H_v$ for $v=\frac{x-y}{||x-y||}$ maps $x$ to $y$.
\[H_vx=y\]

Proposition 2.2.4  Each $A\in\mathbb{R}^{m\times n}$ has a QR decomposition.

Algorithm 2.2.5  The Householder QR factorization algorithm can be used to calculate the $QR$ decomposition for a matrix $A=QR$. The following algorithm computes $R$ by leaving the result in place of $A$ and then computes the columns of $Q$ by applying the Householder transformation for each of the standard basis vectors.

2.3 LU DecompositionJ7NT

Definition 2.3.1  The LU decomposition of a square matrix $A\in\mathbb{C}^{n\times n}$ is the lower triangular matrix $L\in\mathbb{C}^{n\times n}$ and the upper triangular matrix $U\in\mathbb{C}^{n\times n}$ such that
\[A=LU\]

Proposition 2.3.2  Gaussian elimination can be used to construct the matrices $L\in\mathbb{C}^{n\times n}$ and $U\in\mathbb{C}^{n\times n}$ in terms of a set of lower triangular matrices $\{L_1^{-1},\dots,L_n^{-1}\}\subset\mathbb{C}^{n\times n}$ such that
[A = LU = Lb v

Proposition 2.3.3  Let $L_k\in\mathbb{n\times n}$ be the row operation matrix defined by
\[L_{k} = \begin{pmatrix}
1&\ &\ &\ &\ &\ \\
\ &\ddots&\ &\ &\ &\ \\
\ &\ &1&\ &\ &\ \\
\ &\ &-\ell_{k+1,k}&\ddots &\ &\ \\
\ &\ &\vdots&\ &\ddots &\ \\
\ &\ &-\ell_{n,k}&\ &\ &1
\end{pmatrix}\]
where $\ell_{j,k} = \frac{a_{j,k}}{a_{k,k}}$ for $j = k+1,\dots,n$. The inverse $L_k^{-1}$ is the matrix
\[L_{k} = \begin{pmatrix}
1&\ &\ &\ &\ &\ \\
\ &\ddots&\ &\ &\ &\ \\
\ &\ &1&\ &\ &\ \\
\ &\ &\ell_{k+1,k}&\ddots &\ &\ \\
\ &\ &\vdots&\ &\ddots &\ \\
\ &\ &\ell_{n,k}&\ &\ &1
\end{pmatrix}\]

Algorithm 2.3.4  The LU decomposition algorithm without pivoting can be used to compute the LU decomposition of a matrix $A$ by using Gaussian elimination.

Algorithm 2.3.5  The LU decomposition algorithm with partial pivoting can be used to compute the LU decomposition of a matrix $A$ by using Gaussian elimination in a more stable way than without pivoting.

2.4 Cholesky Decomposition8J25

Definition 2.4.1  A positive definite matrix $A\in\mathbb{C}^{n\times n}$ is a Hermitian matrix such that $x^*Ax\geq 0,\ \forall x\in\mathbb{C}^n$ and "$=$" holds only when $x=0$.

Proposition 2.4.2  If a matrix $A\in\mathbb{C}^{n\times n}$ is positive definite, then the block diagonal sub-matrices are also positive definite.

Definition 2.4.3  The Cholesky decomposition of a positive definite matrix $A\in\mathbb{C}^{n\times n}$ is an upper triangular matrix $R\in\mathbb{C}^{n\times n}$ with positive diagonal elements such that
\[ A = R^*R\]

Theorem 2.4.4  Any positive definite matrix has a Cholesky decomposition.

Algorithm 2.4.5  The Cholesky decomposition algorithm can be used to compute the Cholesky decomposition of a matrix $A\in\mathbb{C}^{n\times n}$.

3 Eigenvalue Problems6W74

3.1 Rayleigh Quotient and Inverse IterationWPFA

Proposition 3.1.1  Let $|\lambda_1| > |\lambda_2| > \dots > |\lambda_n|$ be the eigenvalues of a real matrix $A\in\mathbb{R}^{n\times n}$ with corresponding eigenbasis $q_1,q_2,\dots,q_n\in\mathbb{R}^n$. If $v^{(0)}\in\mathbb{R}^n$ is some vector such that $\langle q, v^{(0)}\rangle \neq 0$, then
\[v^{(k)} = \frac{A^kv^{(0)}}{||A^kv^{(0)}||} \to \pm q_1\quad \text{ as } k\to \infty\]

Definition 3.1.2  The Rayleigh quotient of a symmetric matrix $A\in\mathbb{R}^{n\times n}$ is the function $r_A:\mathbb{R}^{n} - \{0\} \to \mathbb{R}$ defined by $r_A(x) = \frac{x^T Ax}{||x||^2}$.

Proposition 3.1.3  A nonzero vector $x\in\mathbb{R}^n$ is an eigenvector of a symmetric matrix $A\in\mathbb{R}^{n\times n}$ if and only if $x$ is a critical point of $r_A(x)$.

Algorithm 3.1.4  The power iteration algorithm can be used to approximate the largest absolute eigenvalue and corresponding eigenbases vector of a matrix $A$.

Algorithm 3.1.5  The Inverse iteration algorithm can be used to

Algorithm 3.1.6  The Rayleigh quotient iteration algorithm can be used to approximate all the eigenvalue and corresponding eigenbases vectors of a matrix $A$.

3.2 Schur Decomposition5JKD

Definition 3.2.1  A similar matrix $A\in\mathbb{C}^{n\times n}$ to another matrix $B\in\mathbb{C}^{n\times n}$ is a matrix where there exists a non-singular matrix $S\in\mathbb{C}^{n\times n}$ such that $A = S^{-1}BS$.

Definition 3.2.2  The singularity transformation for a non-singular matrix $S\in\mathbb{C}^{n\times n}$ is the mapping $\text{Sim}_S : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ defined by
\[\text{Sim}_S(A) = S^{-1}AS\]

Definition 3.2.3  A matrix $A\in\mathbb{C}^{n\times n}$ is diagonalizable iff there exists a non-singular $S\in\mathbb{C}^{n\times n}$ such that $S^{-1}AS$ is a diagonal matrix.

Definition 3.2.4  A matrix $A\in\mathbb{C}^{n\times n}$ is unitary diagonalizable iff there exists a unitary $U\in\mathbb{C}^{n\times n}$ such that $U^* AU$ is a diagonal matrix.

Definition 3.2.5  A matrix $A\in\mathbb{C}^{n\times n}$ is unitary triangularizable iff there exists a unitary $U\in\mathbb{C}^{n\times n}$ such that $U^* AU$ is upper triangular matrix.

Definition 3.2.6  The Schur decomposition of $A\in\mathbb{C}^{n\times n}$ is a unitary Q and upper triangular T such that
\[A = QTQ^*\]

Proposition 3.2.7  A matrix is diagonalizable iff A has n eignenvectors that form a basis of $\mathbb{C}^n$.

Proposition 3.2.8  A matrix is unitary diagonalizable iff $A^*A = AA^*$

Proposition 3.2.9  Any matrix $A\in\mathbb{C}^{n\times n}$ has a Schur decomposition $A = QTA^*$.

Proposition 3.2.10  For $A\in\mathbb{C}^{n\times n}$ let $\{A_k\}$ be the sequence of matrices defined by $A_0= A$, $A_{k-1} = Q_kR_k$, and $A_k = R_k Q_k$, then

1. $A_k$ has the same eigenvalues of $A$.
2. $A^{k} = (Q_1\dots Q_k)(R_k\dots R_1)$.

Theorem 3.2.11  If $A\in\mathbb{C}^{n\times n}$ is a matrix with eigenvalues $\lambda_1,\dots,\lambda_n$ such that $|\lambda_1| > \dots > |\lambda_n| > 0$, then the sequence of matrices $\{T_k\}$ defined by $T_0= A$, $T_{k-1} = Q_kR_k$, and $T_k = R_k Q_k$ converges to $T$ and $Q_1Q_2\dots$ converges to $Q$ in the Schur decomposition $A = QTQ^*$.

Algorithm 3.2.12  The pure QR Schur decomposition algorithm can be used to iteratively compute the Schur decomposition of a matrix $A=QTQ^*$.

Def 3.2.13  An matrix $H\in\mathbb{C}^{n\times n}$ is upper Hessenberg iff $H_{i,j}=0$ for $i>j+1$.

Proposition 3.2.14  All square matrices are unitarily similar to an upper hessenber matrix.

Algorithm 3.2.15  The practical QR Schur decomposition algorithm can be used to iteratively compute the Schur decomposition of a matrix $A=QTQ^*$ more efficiently.

3.3 Arnoldi and Lanczos IterationFH81

Definition 3.3.1  the Krylov subspace of order r generated by $A\in\mathbb{C}^{n\times n}$ and $b\in\mathbb{C}^n$ is the subspace $\mathcal{K}_r(A,b)$ defined by
\[\mathcal{K}_r(A,b) = \text{span}\{b,Ab,A^2b,\dots,A^{r-1}b\}\]

Definition 3.3.2  The Krylov matrix of order r generated by $A\in\mathbb{C}^{n\times n}$ and $b\in\mathbb{C}^n$ is the matrix $K_r(A,b)$ defined by
\[K_r(A,b) = \text{matrix with columns }(b,Ab,A^2b,\dots,A^{r-1}b)\]

Definition 3.3.3  A monic polynomial of degree d is a function $p:\mathbb{C}\to\mathbb{C}$ of the form
\[p(t) = t^d + c_{d-1}t^{d-1} + \dots c_{1}t^{1} + c_0\]

Definition 3.3.4  The minimal polynomial of A with respect to b is the non-zero monic polynomial of lowest degree such that $p(A)b=0$.

Proposition 3.3.5  Let $m$ be the degree of the minimal polynomial of $A$ with respect to $b$, then

• $\{b,Ab,\dots,A^{r-1}b\}$ is linearly independent for $r\leq m$.
• $A^mb,A^{m+1}b,\dots \in \mathcal{K}_m(A,b)$.

Algorithm 3.3.6  The Arnoldi iteration algorithm can be used to find an orthonormal eigenbasis of a matrix $A\in\mathbb{C}^{n\times n}$.

Algorithm 3.3.7  The Lanczos iteration algorithm can be used to find an orthonormal eigenbasis of a matrix $A\in\mathbb{C}^{n\times n}$.

Proposition 4  The projections $\text{Proj}_{W}(v),\text{Proj}_{W^\perp}(v)$ of $\mathbf{v}\in \mathbb{C}^n$ onto $W,W^\perp\subset\mathbb{C}^n$ are orthogonal.

Linear AlgebraTF4T

1 Fundamental Linear AlgebraZ6PT

1.1 Vector SpacesD28A

Definition 1.1.1  A Vector Space is a set $V$ equipped with addition and scalar multiplication with the following properties

1. Commutativity: $\mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v}$, $\forall\ \mathbf{v},\mathbf{w}\in V$
2. Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w})$, $\forall\ \mathbf{u},\mathbf{v},\mathbf{w}\in V$
3. Zero Vector: $\exists\ \mathbf{0}\in V$ such that $\mathbf{v}+\mathbf{0}=\mathbf{v}$, $\forall\ \mathbf{v}\in V$
4. Additive inverse: For every $\mathbf{v}\in V$, there exists $-\mathbf{v}\in V$ such that $\mathbf{v}-\mathbf{v}=\mathbf{0}$
5. Multiplicative Identity: $1\mathbf{v} = \mathbf{v}$ for all $\mathbf{v}\in V$
6. Multiplicative Associativity: $(\alpha\beta)\mathbf{v}$ = $\alpha(\beta\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha, \beta$
7. Additive Closure: $\mathbf{v}+\mathbf{w}\in V$ for all $\mathbf{v},\mathbf{w}\in V$
8. Multiplicative Closure: $\alpha\mathbf{v}\in V$ for all $\mathbf{v}\in V$ and scalars $\alpha$
9. $\alpha(\mathbf{u}+\mathbf{v}) = \alpha\mathbf{u} + \alpha\mathbf{v}$ for all $\mathbf{u},\mathbf{v}\in V$ and scalars $\alpha$
10. $(\alpha + \beta)\mathbf{v}=\alpha\mathbf{v}+\beta\mathbf{v}$ for all $\mathbf{v}\in V$ and scalars $\alpha, \beta$

Proposition 1.1.2  $\mathbb{C}^n$ is a vector space with scalars in $\mathbb{C}$.

Definition 1.1.3  A subspace of a vector space $V$ is a subset $S\subset V$ that has additive closure and multiplicative closure.

Definition 1.1.4  A linear combination of a subset of vectors $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_n\}\subset V$ for scalars $\{a_1,a_2,\dots,a_n\}$ is the sum
\[a_1\mathbf{v}_1+a_2\mathbf{v}_2 + \dots + a_n\mathbf{v}_n = \sum_{i=1}^na_i\mathbf{v}_i\]

Definition 1.1.5  A linearly independent set of vectors $\{\mathbf{v}_i\}$ is a set of vectors such that only trivial linear combinations produce the zero vector $\mathbf{0}$.

Definition 1.1.6  A linearly dependent set of vectors $\{\mathbf{v}_i\}$ is a set of vectors such that there exists a non-trivial linear combination.

Definition 1.1.7  The span of a set of vectors is the set of all possible linear combinations.

Definition 1.1.8  A basis of a vector space $V$ is a set of vectors with span $V$ that is linearly independent.

Definition 1.1.9  The dimension denoted $\text{Dim}(V)$ of a vector space $V$ is the number of vectors in any basis of $V$.

Definition 1.1.10  The inner product denoted $\mathbf{v}\cdot\mathbf{w}\in\mathbb{C}$ of two vectors $\mathbf{v},\mathbf{w}\in V$ is the sum of the product of there elements.
\[\mathbf{v}\cdot\mathbf{w} = \langle \mathbf{v},\mathbf{w} \rangle = \mathbf{v}^*\mathbf{w} = \sum_i{\bar{v_i}w_i} \in \mathbb{C}\]

Definition 1.1.11  The outer product denoted $\mathbf{v}\mathbf{w}^*\in\mathbb{C}^{n\times n}$ of two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}$ is the matrix of products of there elements.
\[\mathbf{v}\mathbf{w}^* = (v_iw_j) \in \mathbb{C}^{n\times n}\]

Definition 1.1.12  The elementary vectors denoted $\mathbf{e}_i\in\mathbb{C}^n$ are the vectors with $e_i = 1$ and zeros for all other elements.

1.2 MatricesZ05T

Definition 1.2.1  A linear transformation is a function $T:V\to W$ for vector fields $V,W$ such that

1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v})+ T(\mathbf{w})$ for all $\mathbf{v},\mathbf{w}\in V$
2. $T(\alpha\mathbf{v}) = \alpha T(\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha$

Definition 1.2.2  A matrix is an grid of entries denoted as a rectangular array, a series of column vectors or a series of row vectors.

Definition 1.2.3  Matrix-vector multiplication denoted $M\mathbf{v}\in V_m$ for a matrix $M=(\mathbf{t}_1,\mathbf{t}_2,\dots,\mathbf{t}_n)\in M_{m\times n}$ and a vector $\mathbf{v}\in V_n$ is
\[M\mathbf{v} = \sum_{i=1}^{n}{v_i \mathbf{m}_i}\quad\text{where } \mathbf{m}_i \text{ are the column vectors of }T\]

Definition 1.2.4  Matrix multiplication denoted $AB\in M_{m\times n}$ for two matrices $A \in M_{m\times k}$ and $B\in M_{k\times n}$ is the matrix where the elements are the dot products of the row vectors of $A$ and the column vectors of $B$. Let $\mathbf{a}_i$ be the row vectors of $A$ and $\mathbf{b}_j$ be the column vectors of $B$, then the elements of $AB$ matrix are
\[AB_{i,j} = (\mathbf{a}_i\cdot\mathbf{b}_j)\]

Proposition 1.2.5  Any matrix $M\in \mathbb{C}^{m\times n}$ defines a linear transformation $M:\mathbb{C}^n\to\mathbb{C}^m$ defined by matrix vector multiplication.

Proposition 1.2.6  Matrix multiplication is the same as function composition of the corresponding linear transformations.

Definition 1.2.7  The conjugate denoted $\overline{M}$ of a matrix $M=(m_{i,j})\in\mathbb{C}^{m\times n}$ is the matrix with complex conjugate elements of $M$.
\[\overline{M} = (\bar{m}_{i,j})\]

Definition 1.2.8  The transpose denoted $M^T$ of a matrix $M=(m_{i,j})\in\mathbb{C}^{m\times n}$ is the matrix with swapped rows and columns of $M$.
\[M^T = (m_{j,i})\]

Definition 1.2.9  The adjoint denoted $M^*$ of a matrix $M=(m_{ij})\in\mathbb{C}^{n\times m}$ is conjugate transpose of $M$.
\[M^* = \overline{M}^T = (\bar{m}_{ji})\]

Definition 1.2.10  The inner product denoted $\mathbf{v}\cdot\mathbf{w}\in\mathbb{C}$ of two vectors $\mathbf{v},\mathbf{w}\in V$ is the sum of the product of there elements.
\[\mathbf{v}\cdot\mathbf{w} = \langle \mathbf{v},\mathbf{w} \rangle = \mathbf{v}^*\mathbf{w} = \sum_i{\bar{v_i}w_i} \in \mathbb{C}\]

Definition 1.2.11  The outer product denoted $\mathbf{v}\mathbf{w}^*\in\mathbb{C}^{n\times n}$ of two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^{n}$ is the matrix of products of there elements.
\[\mathbf{v}\mathbf{w}^* = (v_iw_j) \in \mathbb{C}^{n\times n}\]

Definition 1.2.12  The elementary vectors denoted $\mathbf{e}_i\in\mathbb{C}^n$ are the vectors with $e_i = 1$ and zeros for all other elements.

Definition 1.2.13  The identity matrix is the matrix $I\in\mathbb{C}^{n\times n}$ such that $\mathbf{v} = I\mathbf{v}$ for all $\mathbf{v}\in \mathbb{C}^n$.
\[I = (\mathbf{e}_1,\mathbf{e}_2,\dots,\mathbf{e}_n)\]

Definition 1.2.14  The Kronecker delta $\delta_{ij} = \left\{\begin{array}{lr}
1, & \text{if } i = j\\
0, & \text{if } i\neq j
\end{array}\right\}$.

Definition 1.2.15  The kernel or null space denoted $\text{Ker}(M)$ of a matrix $M$ is the set of vectors $\{ \mathbf{v}\in V : M\mathbf{v} = \mathbf{0} \}$ that map to zero.

Definition 1.2.16  The nullity denoted $\text{Nullity}(M)$ of a matrix $M$ is the dimension of the kernel.

Definition 1.2.17  The range denoted $\text{Ran}(M)$ of a matrix $M$ is the set of vectors $\{ \mathbf{w}\in W : \exists \mathbf{v}\in V\text{ st. }M\mathbf{v}=\mathbf{w} \}$ in the span of the columns of $M$.

Definition 1.2.18  The rank denoted $\text{Rank}(M)$ of a matrix $M$ is the dimension of the range.

Proposition 1.2.19  The kernel and range of a matrix are subspaces of there respective vector spaces.

Theorem 1.2.20  The rank nullity theorem states that the sum of the rank and the nullity is equal to the number of columns.
\[\text{Rank}(M)+\text{Nullity}(M) = \text{ # of columns of M}\]

Definition 1.2.21  A matrix $M$ is invertible iff there exists a matrix $M^{-1}$ such that $MM^{-1} = M^{-1}M = I$.

Definition 1.2.22  The determinant denoted $\text{det}(A)\in\mathbb{C}$ of a matrix $A\in\mathbb{C}^{n\times n}$ with elements $a_{ij}\in\mathbb{C}$ is
\[\text{det}(A) = \sum_{\sigma\in\text{Perm}(\{1,2,\dots, n\})} a_{\sigma(1),1}a_{\sigma(2),2},\dots a_{\sigma(n),n}(-1)^{K(\sigma)}\]

Theorem 1.2.23  For any $A\in\mathbb{C}^{n\times n}$, the following are equivalent:

1. $A$ is invertible.
2. $\text{Rank}(A) = n$
3. $\text{Range}(A) = \mathbb{C}^n$
4. $\text{Ker}(A) = \emptyset$
5. $\text{det}(A)\neq 0$

Definition 1.3  A linear transformation is a function $T:V\to W$ for vector fields $V,W$ such that

1. $T(\mathbf{v}+\mathbf{w}) = T(\mathbf{v})+ T(\mathbf{w})$ for all $\mathbf{v},\mathbf{w}\in V$
2. $T(\alpha\mathbf{v}) = \alpha T(\mathbf{v})$ for all $\mathbf{v}\in V$ and scalars $\alpha$

1.4 Orthogonal and Orthonormal VectorsP8NF

Definition 1.4.1  The 2-norm denoted $||\mathbf{v}||$ of a vector $\mathbf{v}\in\mathbb{C}^n$ is the real number $||\mathbf{v}|| = \sqrt{\langle \mathbf{v}, \mathbf{v}\rangle}$

Definition 1.4.2  Two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ are orthogonal iff $\langle \mathbf{v},\mathbf{w}\rangle = 0$.

Definition 1.4.3  Two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ are orthonormal iff they are orthogonal and $||\mathbf{v}||=||\mathbf{w}||=1$.

Definition 1.4.4  The distance between two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ is the 2-norm of their difference $||\mathbf{v}-\mathbf{w}||$.

Definition 1.4.5  The angle between two vectors $\mathbf{v},\mathbf{w}\in\mathbb{C}^n$ is $\arccos\frac{\langle \mathbf{v},\mathbf{w} \rangle}{||\mathbf{v}||\cdot||\mathbf{w}||}$.

Definition 1.4.6  The orthogonal subspace denoted $W^{\perp}\subset \mathbb{C}^n$ of a subset $W\subset\mathbb{C}^n$ is the subspace of vectors that are orthogonal to $W$.

Proposition 1.4.7  Orthogonal vectors are linearly independent.

Proposition 1.4.8  If $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v})n\}\subset\mathbb{C}^n$ is orthonormal then any vector $\mathbf{v}\in\mathbb{C}^n$ can be written as $\mathbf{v} = \langle \mathbf{v}_1, \mathbf{v}\rangle\mathbf{v}_1 + \langle \mathbf{v}_2, \mathbf{v}\rangle\mathbf{v}_2 + \dots + \langle \mathbf{v}_n, \mathbf{v}\rangle\mathbf{v}_n$.

Proposition 1.4.9  If $\{\mathbf{v}_1,\mathbf{v}_2,\dots\}$ is a basis of a subspace $W\subset \mathbb{C}^n$ and $\{\mathbf{w}_1,\mathbf{w}_2,\dots\}$ is a basis of $W^\perp\subset \mathbb{C}^n$, then $\{\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{w}_1,\mathbf{w}_2,\dots\}$ is a basis of a $\mathbb{C}^n$ and $\dim(W)+\dim(W^{\perp}) = n$.

Proposition 1.4.10  For any matrix $A\in\mathbb{C}^{m\times n}$,

1. $\text{Ker}(A)^\perp = \text{Ran}(A^*)$
2. $\text{Rang}(A)^\perp = \text{Ken}(A^*)$

1.5 Hermitian and Unitary Matrices1JJJ

Definition 1.5.1  A matrix $A\in\mathbb{C}^{n\times n}$ is Hermitian iff $A^*=A$.

Definition 1.5.2  A matrix $A\in\mathbb{C}^{n\times n}$ is unitary iff $A^*A=AA^*=I$.

Proposition 1.5.3  If $Q\in\mathbb{C}^{n\times n}$ is unitary, then

1. $||Q\mathbf{u}||=||\mathbf{u}||, \quad\forall \mathbf{u}\in\mathbb{C}^n$.
2. The rows and columns of $Q$ are an orthonormal basis of $\mathbb{C}^n$.

Theorem 1.5.4  Eigenvalue Decomposition of Hermitian Matrices Theorem states that any Hermitian matrix $A\in\mathbb{C}^{n\times n}$ can be diagonalized by an Hermitian matrix $Q\in\mathbb{C}^{n\times n}$ that is
\[A = Q\begin{pmatrix}\lambda_1 && \ &&\ \\ \ && \ddots&&\ \\\ &&\ &&\lambda_n\end{pmatrix}Q^*\]
where $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $A$ and the columns of $Q$ are eigenvectors of $A$.

1.6 Singular Value DecompositionM6C4

Definition 1.6.1  A singular value of a matrix $A\in\mathbb{C}^{n\times m}$ is a real number $\sigma > 0$ such that $\sigma^2$ is an eigenvalue of $A^*A$ or $AA^*$.

Theorem 1.6.2  Full Singular Value Decomposition Theorem states that for any $A\in\mathbb{C}^{n\times m}$, there exists unitary $U\in\mathbb{C}^{n\times n}$, unitary $V\in\mathbb{C}^{m\times m}$ and semi-diagonal $\Sigma\in\mathbb{C}^{n\times m}$ such that
\[A = U\Sigma V^*\]
where the first $r$ diagonal elements of $\Sigma$ are the singular values $\sigma_1,\dots,\sigma_r$ of $A$ and the remaining elements of $\Sigma$ are zero.

Theorem 1.6.3  Reduced Singular Value Decomposition Theorem states that for any $A\in\mathbb{C}^{n\times m}$, there exists $U\in\mathbb{C}^{n\times r}$, unitary $V\in\mathbb{C}^{r\times m}$ and diagonal $\Sigma\in\mathbb{C}^{r\times r}$ such that
\[A = U\Sigma V^*\]
where the diagonal elements of $\Sigma$ are the singular values $\sigma_1,\dots,\sigma_r$ of $A$.

Proposition 1.6.4  For any matrix $A\in\mathbb{C}^{n\times m}$, the eigenvalue decomposition of $A^*A = U(\Sigma^*\Sigma)U^*$ and the eigenvalue decomposition $AA^*=V(\Sigma\Sigma^*)V^*$ determine the singular value decomposition $A=U\Sigma V^*$.

Proposition 1.6.5  The number of singular values of a matrix is the rank.

Proposition 1.6.6  For any matrix $A\in\mathbb{C}^{n\times m}$ with full singular value decomposition $A=U\Sigma V^*$ and any vector $x\in\mathbb{C}^m$
\[Ax = \sum_{j=1}^r{\sigma_j\langle v_j,x\rangle}u_j\]
where $\sigma_1,\dots,\sigma_r$ are the singular values of $A$, $v_j$ are the columns of $V$ and $u_j$ are the columns of $U$.

1.7 NormsHNPM

Definition 1.7.1  A norm is a function $||\cdot||:V\to[0,\infty)$ with the following properties:

1. Positive definiteness: $||v||\geq0$, $\forall v\in V$
2. Scaling: $||cv||=|c||||v|$, $\forall v\in V$ and scalars $c$
3. Trianglular inequality: $||u+v||\leq ||u||+||v||$, $\forall u,v\in V$

Definition 1.7.2  The p-norm denoted $||\mathbf{v}||_p$ of a vector $\mathbf{v}\in\mathbb{C}^n$ for $1\leq p <\infty$ is the real number $||\mathbf{v}||_p = \left(\sum_{i=1}^n{|v_i^p|}\right)^{\frac{1}{p}}$.

Definition 1.7.3  The $\infty$-norm denoted $||\mathbf{v}||_\infty$ of a vector $\mathbf{v}\in\mathbb{C}^n$ is the real number $||\mathbf{v}||_\infty = \max_{1\leq j \leq n}|v_j|$.

1.7.4 

Proposition 1.7.5  Holder's Inequality states that for $1\leq p, q < \infty$ such that $\frac{1}{p}+\frac{1}{q}=1$, then for any $x,y\in\mathbb{C}^n$,
\[|x\cdot y|\leq||x||_p||y||_q\]

1.8 Matrix Norms1AHP

Definition 1.8.1  The matrix norm denoted $||A||$ of a matrix $A\in\mathbb{C}^{m\times n}$ induced by a vector norm $||\cdot||$ is the real number $||A|| = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||}{||x||}=\max_{||x||=1}||Ax||$

Definition 1.8.2  The matrix p-q-norm denoted $||A||_{p,q}$ of a matrix $A\in\mathbb{C}^{m\times n}$ for $1\leq p,q \leq \infty$ is the real number $||A||_{p,q} = \max_{x\in \mathbb{C}^m/\{0\}}\frac{||Ax||_p}{||x||_q}=\max_{||x||_q=1}||Ax||_p$

Proposition 1.8.3  For two matrices $A\in\mathbb{C}^{m\times k}$ and $B\in\mathbb{C}^{k\times n}$, the following inequality holds for any $1\leq p,q,r\leq \infty$.
\[||AB||_{p,r}\leq ||A||_{p,q}||B||_{q,r}\]

Proposition 1.8.4  The matrix $1$-norm is the max of the column sums. For any matrix $A\in\mathbb{C}^{m\times n}$ with column vectors $\{\mathbf{c}_1,\dots,\mathbf{c}_n\}$, $||A||_1 = \max_{j\in \{1,\dots,n\}}||\mathbf{c}_j||_1$.

Proposition 1.8.5  Matrix multiplication by unitary matrices preserves $2$-norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_2 = ||AV||_2 = ||A||_2\]

Definition 1.8.6  The Frobenius norm denoted $||A||_F$ of a matrix $A\in\mathbb{C}^{m\times n}$ is the real number $||A||_F = \sqrt{\text{Tr}(A^*A)} = \sqrt{\sum_{i=1}^m{\sum_{j=1}^n{|A_{i,j}|^2}}}$.

Proposition 1.8.7  Matrix multiplication by unitary matrices preserves Frobenius norms. For any $A\in\mathbb{C}^{m\times n}$ and unitary $U\in\mathbb{C}^{m\times m}$, $V\in\mathbb{C}^{n\times n}$,
\[||UA||_F = ||AV||_F = ||A||_F\]

Corollary 1.8.8  For $A\in\mathbb{C}^{m\times n}$ with singular values $\sigma_1,\dots,\sigma_r$, $||A||_F = \sqrt{\sigma_1^2+\dots+\sigma_r^2}$

1.8.9 

1.9 Orthogonal ProjectorsFZ2N

Definition 1.9.1  The projection denoted $\text{Proj}_W(\mathbf{v})\in W$ of a vector $\mathbf{v}\in\mathbb{C}^n$ onto a subspace $W\subset \mathbb{C}^n$ is defined for an orthonormal basis $\{\mathbf{u}_1,\dots\mathbf{u}_r\}$ of $W$ by
\[\text{Proj}_W(\mathbf{v}) = \langle \mathbf{u}_1,\mathbf{v} \rangle \mathbf{v} + \dots + \langle \mathbf{u}_r,\mathbf{v} \rangle \mathbf{v} \]

Proposition 1.9.2  The projections $\text{Proj}_{W}(v),\text{Proj}_{W^\perp}(v)$ of $\mathbf{v}\in \mathbb{C}^n$ onto $W,W^\perp\subset\mathbb{C}^n$ are orthogonal.

Definition 1.9.3  An orthogonal projector matrix is a matrix $P\in\mathbb{C}^{n\times n}$ such that $P^2 = P^* = P$.

Proposition 1.9.4  If $P$ is an orthogonal projector, then $I-P$ is also an orthogonal projector.

Proposition 1.9.5  The product $\mathbf{u}\mathbf{u}^*$ for $\mathbf{u}\in\mathbb{C}^n$ such that $||\mathbf{u}||_2=1$ is an orthogonal projector that projects onto the span of $\mathbf{u}$.

Proposition 1.9.6  If $W\subset \mathbb{C}^n$ is a subspace with an orthonormal basis $\{\mathbf{u}_1,\dots,\mathbf{u}_k\}$, then the matrix $P=\mathbf{u}_1\mathbf{u}_1^*+\dots+\mathbf{u}_k\mathbf{u}_k^*$ is the orthogonal projector $\text{Proj}_W$ and $\text{Ran}(P)=W$.

Proposition 1.9.7  If $P\mathbf{v}=\text{Proj}_W(\mathbf{v})\ \forall \mathbf{v}\in\mathbb{C}^n$, then $(I-P)\mathbf{v} = \text{Proj}_{W^\perp}(\mathbf{v})$.

Proposition 1.9.8  If $P\mathbf{v}=\text{Proj}_W(\mathbf{v})\ \forall \mathbf{v}\in\mathbb{C}^n$, then $(I-P)\mathbf{v} = \text{Proj}_{W^\perp}(\mathbf{v})$.

1.9.9 

Real Analysis

1.9.10 

Complex Analysis

1.9.11 

Abstract Algebra

1.9.12 

Fourier Analysis

1.9.13 

Topology