# 0000 # Physics # Many Body Physics 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 Fields ### 1.1 Introduction **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_{i1: \[\alpha = \frac{\mu_0 e^2 c}{2 h} = \frac{e^2}{2\varepsilon_0 h c} \approx 0.0072973525643 \approx 1/137.035999177\] **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 Excitation **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. Band Structure of GaAs **File 1.2.4 **  Band_Structure_GaAs.svg **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. | Particle | Location | Effective Mass | | -------- | -------- | ------- | | Electron | $\Gamma$-valley | $m^*_{\Gamma e} \approx 0.067 m_e$ | | Electron | L-valley | $m^*_{Le} \approx 0.85 m_e$ | | Electron | X-valley | $m^*_{Xe} \approx 0.85 m_e$ | | Electron-Hole | Heavy Valence | $m^*_{hh} \approx 0.15 m_e$ | | Electron-Hole | Light Valence | $m^*_{lh} \approx 0.082 m_e$ | **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](https://kaedon.net/l/^60n0#a9cz) 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. Band Structure of GaAs Ahmed, I., et al., Light Science & Applications 10(1), 174. (2021) [10.1038/s41377-021-00609-3](https://kaedon.net/l/^jn26) **File 1.2.10 **  TimelineOfAPulse.png **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](https://kaedon.nethttps://kaedon.net/l/rpch#hnk8). **Definition 1.2.13 **  The **electron-phonon scattering** process is when electrons scatter off lattice vibrations ([phonons](https://kaedon.net/l/^rpch#kwhn)) until they reach [thermal equilibrium](https://kaedon.net/l/^rpch#at50) 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 Effects **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 Lasers **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 Optics ### 2.1 The Wave Equation from Maxwell's Equations **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 Polarization **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](https://kaedon.net/l/^ahc1#wr8f) 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](https://kaedon.net/l/^ccp3) $\vec{H}$. \[\vec{M} = \chi_m\vec{H}\] **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](https://kaedon.net/l/^2anz), $\varepsilon_0$ 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 $\tilde{n}$. | | Dielectric Function | Conductivity | Index of Refraction | | -------- | -------- | ------- | ------- | | $\tilde{\varepsilon}$ | $\tilde{\varepsilon} = \varepsilon_1 + i \varepsilon_2$ | $\varepsilon_1 = \varepsilon_\infty - \sigma_2/\omega,\quad$ $\varepsilon_2=\sigma_1/\omega$ | $\varepsilon_{1}= \varepsilon_0(n^2 - k^2),\quad$ $\varepsilon_2 = 2\varepsilon_0nk$ | | $\tilde{\sigma}$ | | 19.26mm | 32.74mm | | $\tilde{n}$ | 515nm | 19.26mm | 32.74mm | ### 2.3 Nonlinear Suseptability in 1D **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](https://kaedon.net/l/^6re4) into the [wave equation from polarization](https://kaedon.net/l/^60n0#tj13). 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](https://kaedon.net/l/^60n0#391d) for a material with a non-zero $\chi_e^{(3)}$. ### 2.4 Lorentz and Drude Model **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](https://kaedon.net/l/^60n0#j2fk), 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 Matching **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](https://kaedon.net/l/^6re4) into the [wave equation from polarization](https://kaedon.net/l/^60n0#tj13). 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](https://kaedon.net/l/^6ken) 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](https://kaedon.net/l/^tdzf). \[\frac{\partial^2 E_3(z)}{\partial z^2} \pm 2ik_3 \frac{\partial E_3(z)}{\partial z} = \frac{-4d_{eff}\omega_3^2}{c^2}E_1E_2e^{\pm i(k_1+k_2 - k_3)z}\] **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](https://kaedon.net/l/^60n0#jk1j) is made for the [differential equation for 1D sum frequency generation](https://kaedon.net/l/^60n0#817f), 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](https://kaedon.net/l/^60n0#jk1j) 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](https://kaedon.net/l/^60n0#jk1j) 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 Birefringence **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 Matching **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 Matching **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 Amplifiers **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 Scattering **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 **  Types of Ramen Scattering **File 2.10.5 **  Ramanscattering.svg **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](https://kaedon.net/l/^h407) 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](https://kaedon.net/l/^60n0#0w2j) at high intensity. ## 3 Ultrafast Lasers ### 3.1 Keys to Laser Operation **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 Cavities **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 Mechanics $\newcommand{\dj}{\rlap{d}{\bar{\phantom{w}}}}$ ## 1 Thermodynamics ### 1.1 1st Law of Thermodynamics **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 Thermodynamics **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 Systems **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 Enthalpy **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 Energy **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 Energy **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 Potential **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 Thermodynamics **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 Mechanics ### 2.1 Microcanonical Ensemble **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 Ensemble **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 Ensemble **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 Ensemble **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 Mechanics **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 Mechanics **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 Gases ### 3.1 Identical Particles **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 Ensemble **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](#99eh). \[\ell_Q^2 >> \left(\frac{V}{N}\right)^{2/3}\] ### 3.3 Quantum Gases in the Grand Canonical Ensemble **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 States **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 Gases **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](https://kaedon.net/l/0j0H) 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 Gases **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 Condensate **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](https://kaedon.net/l/raa2). \[P \approx f_{5/2}^{+1}(1) \frac{k_B T}{\ell_Q^3}\] **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 Gas **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 Gas **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 Transitions ### 4.1 Virial Expansion **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> 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 Transistions **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 Theory **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 \[q_1 = q_3 = \dots = 0\] 3. *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** # [Quantum Mechanics 1](https://dracentis.github.io/pdfs/QuantumMechanics.pdf)

# [Quantum Mechanics 2](https://dracentis.github.io/pdfs/QuantumMechanics2.pdf) # Electromagnetism # Electromagnetism in Free Space ## 1 Introduction ### 1.1 Notation **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 $\hat{x},\hat{y},\hat{z}$, \[\vec{r}=(x,y,z)=x \hat{x} + y \hat{y} + z \hat{z}\] Cartesian Diagram Image **File 1.1.1 **  Cartesian.png **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 $\hat{r},\hat{\theta},\hat{\phi}$, Spherical Diagram Image
| | | |:----------|:-------------| | $r = \sqrt{x^2+y^2+z^2}\quad$ | $\hat{r} = \sin(\theta)\cos(\phi)\hat{x} + \sin(\theta)\sin(\phi)\hat{y}+\cos(\theta)\hat{z}$ | | $\theta = \tan^{-1}(\sqrt{x^2+y^2}/2)\quad$ | $\hat{\theta}=\cos(\theta)\cos(\phi)\hat{x}+\cos(\theta)\sin(\phi)\hat{y}-\sin(\theta)\hat{z}$ | | $\phi = \tan^{-1}(y/x)\quad$ | $\hat{\phi}=-\sin(\phi)\hat{x}+\cos(\phi)\hat{y}$ | | $\ $ | $\ $ | | $x = r\sin\theta\cos\phi\quad$ | $\hat{x} = \sin(\theta)\cos(\phi)\hat{r} + \cos(\theta)\cos(\phi)\hat{\theta} - \sin(\phi)\hat{\phi}$ | | $y = r\sin\theta\sin\phi\quad$ | $\hat{y} = \sin(\theta)\sin(\phi)\hat{r} + \cos(\theta)\sin(\phi)\hat{\theta} + \cos(\phi)\hat{\phi}$ | | $z = r\cos\theta\quad$ | $\hat{z} = \cos(\theta)\hat{r} - \sin(\theta)\hat{\theta}$ |
**File 1.1.2 **  Spherical.PNG **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 $\hat{s},\hat{\phi},\hat{z}$, Cylindrical Diagram Image
| | | |:--|:--| | $s = \sqrt{x^2+y^2}\quad$ | $\hat{s} = \cos(\phi)\hat{x} + \sin(\phi)\hat{y}$ | | $\phi = \tan^{-1}(y/x)\quad$ | $\hat{\phi} = -\sin(\phi)\hat{x} + \cos(\phi)\hat{y}$ | | $z=z\quad$ | $\hat{z}=\hat{z}$ | | $\ $ | $\ $ | | $x = s\cos\phi\quad$ | $\hat{x} = \cos(\phi)\hat{s} - \sin(\phi)\hat{\phi}$ | | $y = s\sin\phi\quad$ | $\hat{y} = \sin(\phi)\hat{s} + \cos(\phi)\hat{\phi}$ | | $z=z\quad$ | $\hat{z}=\hat{z}$ |
**File 1.1.3 **  Cylindrical.PNG **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 Units **Definition 1.2.1 **  The **SI unit system** is the most popular system of units that uses the fundamental units of [seconds](https://kaedon.net/l/^ahc1#h5t8), [meters](https://kaedon.net/l/^ahc1#8dza), [kilograms](https://kaedon.net/l/^ahc1#wahe), [ampere](https://kaedon.net/l/^ahc1#613r) and [Kelvin](https://kaedon.net/l/^ahc1#ncp0) 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 Equations **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 Electrostatics ### 2.1 Electric Field **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](https://kaedon.net/l/^ahc1#wr8f) 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 Potential **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 Harmonics **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 Expansion **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 Variables **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](https://kaedon.net/l/^pd4j#59d9). 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 Variables **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](https://kaedon.net/l/^ahc1#85ep) $P_\ell$ 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 Variables **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 Relaxation **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. Spherical Diagram Image **File 2.8.2 **  Relaxation.PNG ### 2.9 Finite Element Method **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 Theory **Definition 2.10.1 **  A **linear differential operator** is a functional operator involving differentiation that is [linear](https://kaedon.net/l/^n96p). **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](https://kaedon.net/l/^wna1). \[\mathcal{L}G(\vec{r},\vec{r}')=-4\pi\delta(\vec{r}-\vec{r}')\] **Corollary 2.10.3 **  Let $G(\vec{r},\vec{r}')$ be a [Green's function](https://kaedon.net/l/^1mfe) for a [linear differential operator](https://kaedon.net/l/^3r0m) $\mathcal{L}$ 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](https://kaedon.net/l/^fh5f) $\mathcal{L}u(\vec{r})=f(\vec{r})$ is the integral of the [Green's function](https://kaedon.net/l/^1mfe) $G$ of the [linear differential operator](https://kaedon.net/l/3r0m) $\mathcal{L}$ 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](https://kaedon.net/l/^1mfe) for the [Laplace operator](https://kaedon.net/l/^59d9)** $G(\vec{r},\vec{r}')$ 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](https://kaedon.net/l/1mfe) $G$ 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](https://kaedon.net/l/1mfe) $G$ 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](https://kaedon.net/l/^ahc1#1mfe) $G(\vec{r},\vec{r}')$ 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 Magnetostatics ### 3.1 Magnetic Field **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](https://kaedon.net/l/7ahh) and [Maxwell's equations](https://kaedon.net/l/3a2h). \[\oint_C\vec{B}\cdot d\vec{\ell} = \mu_0\int_S\vec{J}\cdot d\vec{S} = \mu_0I_{\text{enclosed}}\] **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 Forces **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 Potential **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 Moment **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 Field **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 Relativity ### 4.1 Lorentz Transformations **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](https://kaedon.net/l/^ahc1#awac). **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](https://kaedon.net/l/^ahc1#7823) 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](https://kaedon.net/l/^ahc1#59kr). 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](https://kaedon.net/l/^ahc1#59kr) are [Lorentz transformations](https://kaedon.net/l/^ahc1#7823). **Definition 4.1.9 **  The **Lortenz boost** $A\mapsto A'$ is a [homogeneous Lorentz transformation](https://kaedon.net/l/^ahc1#59kr) that transforms any [four vector](https://kaedon.net/l/^ahc1#m43n) $A\in\mathbb{R}^{1,3}$ to a four vector in a reference frame moving with relative velocity $\vec{v}$ and [simultaneity beta](https://kaedon.net/l/^ahc1#awac) $\vec\beta = \frac{\vec{v}}{c}$. \[A_0' = \gamma(A_0 - \vec{\beta}\cdot\vec{A})\] \[\vec{A}' = \vec{A}+\frac{(\gamma - 1)}{\beta^2}(\vec{\beta}\cdot\vec{A})\vec{\beta}-\gamma\vec{\beta}A_0\] **Result 4.1.10 **  The [Lorentz boost](https://kaedon.net/l/^ahc1#5zte) $A\mapsto A'$ into a frame with relative velocity $\vec{v}$ and [simultaneity beta](https://kaedon.net/l/^ahc1#awac) $\vec\beta = \frac{\vec{v}}{c}$ 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](https://kaedon.net/l/^ahc1#5zte) $A\mapsto A'=\Lambda(\vec{\beta}) A$ to a four vector $A\in\mathbb{R}^{1,3}$ into a reference frame with relative velocity $\vec{v}$ and [simultaneity beta](https://kaedon.net/l/^ahc1#awac) $\vec\beta = \frac{\vec{v}}{c}$. \[\Lambda(\vec{\beta}) = \begin{pmatrix} \gamma & -\gamma\beta_x & -\gamma\beta_y & -\gamma\beta_z \\ -\gamma\beta_x & 1+\frac{(\gamma-1)\beta_x^2}{\beta^2} & \frac{(\gamma-1\beta_x\beta_y)}{\beta^2} & \frac{(\gamma-1\beta_x\beta_z)}{\beta^2}\\ -\gamma\beta_y & \frac{(\gamma-1\beta_x\beta_y)}{\beta^2} & 1+\frac{(\gamma-1)\beta_y^2}{\beta^2} & \frac{(\gamma-1\beta_y\beta_z)}{\beta^2}\\ -\gamma\beta_z & \frac{(\gamma-1\beta_x\beta_z)}{\beta^2} & \frac{(\gamma-1\beta_y\beta_z)}{\beta^2} & 1+\frac{(\gamma-1)\beta_z^2}{\beta^2} \end{pmatrix}\] ### 4.2 Four Vectors **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](https://kaedon.net/l/^ahc1#59kr). **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](https://kaedon.net/l/^ahc1#59kr). **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](https://kaedon.net/l/^ahc1#59kr). \[(ct,\vec{r})\cdot(\omega/c,\vec{k}) = \omega-\vec{k}\cdot\vec{r} = \phi\] **Table 4.2.13 **  Four vectors and their corresponding invariants.$\newcommand\abs[1]{\left|#1\right|}$ | Four Vector | | Invarient | | |:----------------|:---------|:-----------|:----------| | 4-Position | $(ct,\vec{r})$ | Spacetime interval | $(ct)^2 - \abs{\vec{r}}^2 = s^2$ | | 4-Velocity| $(\gamma_vc,\gamma_v\vec{v})$ | (no name) | $(\gamma_vc)^2 - \gamma_v^2\abs{\vec{v}}^2$ | | 4-Momentum | $(E/c,\vec{p})$ | Rest mass squared | $(E/c)^2 - \abs{\vec{p}}^2 = m^2c^2$ | | 4-Current Density | $\left(c\rho,\vec{J}\right)$ | (no name) | $(c\rho)^2 - \abs{\vec{J}}^2$ | | 4-Wave Vector | $(\omega/c,\vec{k})$ | Rest mass over hbar squared | $(\omega/c)^2 - \abs{\vec{k}}^2 = (mc/\hbar)^2$ | ### 4.3 Einstein Notation **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](https://kaedon.net/l/^ahc1#pma4). \[a\cdot b = a_\mu b^\mu = \eta_{\mu\nu}a^\nu b^\mu = \sum_{\mu =0}^3\sum_{\nu=0}^3\eta_{\mu\nu}a^\nu b^\mu = a_0b_0 - \vec{a}\cdot\vec{b}\] ### 4.4 Field Strength Tensor **Definition 4.4.1 **  The **field strength tensor** $F$ is a 4-tensor that contains both the [electric field](https://kaedon.net/l/^ahc1#zf8f) and the [magnetic field](https://kaedon.net/l/^ahc1#h3d5) 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](https://kaedon.net/l/^ahc1#059k) $\Lambda(\vec\beta)$ 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](https://kaedon.net/l/^ahc1#awac) $\vec\beta = \frac{\vec{v}}{c}$ 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](https://kaedon.net/l/^ahc1#awac) $\vec\beta = \frac{\vec{v}}{c}$ 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 Electrodynamics ### 5.1 Slowly Varying Fields **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 Time **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](https://kaedon.net/l/^1mfe) for the [Laplace operator](https://kaedon.net/l/^59d9)** $G(\vec{r},\vec{r}',t,t')$ 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 Equations **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](https://kaedon.net/l/^ahc1#zfmr). \[\vec{E}(\vec{r},t) = \vec{E}_{Coulomb}(\vec{r},t) + \vec{E}_{Charge}(\vec{r},t)+\vec{E}_{Current}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\] \[\vec{E}_{Coulomb}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int \rho(\vec{r}',t')\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\] \[\vec{E}_{Charge}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left[ \frac{1}{c}\frac{\partial \rho(\vec{r}',t')}{\partial t'} \right] \frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^2}d^3r'\] \[\vec{E}_{Current}(\vec{r},t)=-\frac{1}{4\pi\varepsilon_0}\int\left[\frac{1}{c^2}\frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\right]\frac{1}{\abs{\vec{r}-\vec{r}'}}d^3r'\] **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](https://kaedon.net/l/^ahc1#zfmr). \[\vec{E}(\vec{r},t) = \vec{E}_{Coulomb}(\vec{r},t)+\vec{E}_{Current1}(\vec{r},t)+\vec{E}_{Current2}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\] \[\vec{E}_{Coulomb}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int \rho(\vec{r}',t')\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\] \[\vec{E}_{Current1}(\vec{r},t)=\frac{1}{4\pi\varepsilon_0c}\int\frac{\left[\vec{J}(\vec{r}',t')\cdot(\vec{r}-\vec{r}')\right](\vec{r}-\vec{r}')+\left[\vec{J}(\vec{r}',t')\times(\vec{r}-\vec{r}')\right]\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^4}d^3r'\] \[\vec{E}_{Current2}(\vec{r},t)=\frac{1}{4\pi\varepsilon_0c^2}\int\frac{\left[\frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\times(\vec{r}-\vec{r}')\right]\times(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\] **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](https://kaedon.net/l/^ahc1#zfmr). \[\vec{B}(\vec{r},t) = \vec{B}_{BiotSavart}(\vec{r},t) + \vec{B}_{Current}(\vec{r},t)\newcommand\abs[1]{\left|#1\right|}\] \[\vec{B}_{BiotSavart}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\vec{J}(\vec{r}',t')\times\left[\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}\right]d^3r'\] \[\vec{B}_{Current}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\left[ \frac{1}{c} \frac{\partial \vec{J}(\vec{r}',t')}{\partial t'}\right]\times\left[\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^2}\right]d^3r'\] **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](https://kaedon.net/l/^ahc1#zfmr).$\newcommand\abs[1]{\left|#1\right|}$ \[\vec{E}(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left(\rho(\vec{r}',t) - \frac{\abs{\vec{r}-\vec{r}'}^2}{2c^2}\frac{\partial^2\rho(\vec{r}',t)}{\partial t^2} + \cdots\right)\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\] \[+\frac{\mu_0}{4\pi}\int\frac{\partial \vec{J}(\vec{r}',t)}{\partial t}\frac{1}{\abs{\vec{r}-\vec{r}'}}+\cdots d^3r'\] \[\vec{B}(\vec{r},t) = \frac{\mu_0}{4\pi}\int\left[\vec{J}(\vec{r}',t)-\frac{\abs{\vec{r}-\vec{r}'}^2}{2c^2}\frac{\partial^2\vec{J}(\vec{r}',t)}{\partial t^2}\right]\times\frac{(\vec{r}-\vec{r}')}{\abs{\vec{r}-\vec{r}'}^3}d^3r'\] ### 5.4 Electromagnetic Energy and Momentum **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](https://kaedon.net/l/^ahc1#059k) $\Lambda(\vec\beta)$ to a 4-tensor $T$. \[T' = \Lambda(\vec{\beta})T\Lambda^\top\] ### 5.5 Multipole Radiation **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](https://kaedon.net/l/^ahc1#pa5a) $\vec{X}_{\ell,m}(\theta,\phi)$. \[\vec{E} = \text{Real}\left( \sqrt{\frac{\mu_0}{\varepsilon_0}} \sum_{\ell,m}\left[ a_E^{\ell,m}\nabla\times\left( \frac{i}{k}f_\ell(kr)\vec{X}_{\ell,m} \right) + a_M^{\ell,m}g_\ell(kr) \vec{X}_{\ell,m} \right] e^{-i\omega t} \right)\] \[\vec{B} = \text{Real}\left( \mu_0\sum_{\ell,m}\left[ -a_E^{\ell,m}\nabla\times\left( \frac{i}{k}g_\ell(kr)\vec{X}_{\ell,m} \right) + a_M^{\ell,m}f_\ell(kr) \vec{X}_{\ell,m} \right] e^{-i\omega t} \right)\] \[f_\ell(kr) = A_\ell^{(1)}H_\ell^{(1)}(kr) + A_\ell^{(2)}H_\ell^{(2)}(kr)\] \[g_\ell(kr) = B_\ell^{(1)}H_\ell^{(1)}(kr) + B_\ell^{(2)}H_\ell^{(2)}(kr)\] \[k = \frac{\omega}{c}\] 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](https://kaedon.net/l/^ahc1#emjk). **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](https://kaedon.net/l/^ahc1#hae2) $Y_{\ell,m}^*$ and [Bessel functions](https://kaedon.net/l/^ahc1#et2j) $J_{\ell + 1/2}$. 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](https://kaedon.net/l/^ahc1#hae2) $Y_{\ell,m}^*$ and [Bessel functions](https://kaedon.net/l/^ahc1#et2j) $J_{\ell + 1/2}$. 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](https://kaedon.net/l/^ahc1#hae2) $Y_{\ell,m}^*$ and [Bessel functions](https://kaedon.net/l/^ahc1#et2j) $J_{\ell + 1/2}$. 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 Charges **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 Appendix ### 6.1 References [^32ca] [^32ca]: Richard S. Davis; Determining the value of the fine-structure constant from a current balance: Getting acquainted with some upcoming changes to the SI. Am. J. Phys. 1 May 2017; 85 (5): 364–368. https://doi.org/10.1119/1.4976701 # Electromagnetism in Materials ## 1 Introduction ### 1.1 Logistics **File 1.1.1 **  syllabus_EM2_2024.pdf ### 1.2 Notation **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 Units **1.3.1 SI Units** **Definition 1.3.2 **  The **SI unit system** is the most popular system of units that uses the fundamental units of [seconds](https://kaedon.net/l/^ahc1#h5t8), [meters](https://kaedon.net/l/^ahc1#8dza), [kilograms](https://kaedon.net/l/^ahc1#wahe), [ampere](https://kaedon.net/l/^ahc1#613r) and [Kelvin](https://kaedon.net/l/^ahc1#ncp0) 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](https://kaedon.net/l/^ahc1#8dza), [grams](https://kaedon.net/l/^ahc1#wahe), [seconds](https://kaedon.net/l/^ahc1#h5t8) and [Kelvin](https://kaedon.net/l/^ahc1#ncp0) 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 Conductors ### 2.1 Microscopic Electric Field **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](https://kaedon.net/l/^ahc1#zf8f) 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<> \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 Friction **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 Equations **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 Diffusion **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 Tunneling **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 Tunneling **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 Transport ### 3.1 Bloch's Theorem and Berry Phase **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 Effect **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 Conductivity **4.1 **  Impurity Scattering **4.2 **  Electric Conductivity **4.3 **  Thermal Conductivity **4.4 **  Magnetoconductivity **4.5 **  Cyclotron Resonance ## 5 Low-dimensional Systems **5.1 **  Landauer 1D Conductivity **5.2 **  2D Electron Systems **5.3 **  Quantum Hall Effect ## 6 Localization **6.1 **  Weak Localization **6.2 **  Anderson Localization **6.3 **  Density Matrix and the Quantum Kinetic Equation ## 7 Phonon **7.1 **  Electron-phonon Interaction **7.2 **  Polaronic Effect **7.3 **  Ohmic Dissipation **7.4 **  The Orthogonality Catastrophe **7.5 **  Holstein Polarons ## 8 Topological Materials **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 # Math # Numerical Linear Algebra ## 1 Fundamental Linear Algebra ### 1.1 Vector Spaces **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 Matrices **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 Vectors **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 Matrices **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 Decomposition **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 Norms **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 Norms **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 Projectors **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 Decomposition ### 2.1 QR Decomposition **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 Decomposition **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 Decomposition **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 Decomposition **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 Problems ### 3.1 Rayleigh Quotient and Inverse Iteration **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 Decomposition **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 Iteration **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 Algebra ## 1 Fundamental Linear Algebra ### 1.1 Vector Spaces **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 Matrices **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 Vectors **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 Matrices **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 Decomposition **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 Norms **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 Norms **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 Projectors **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](https://dracentis.github.io/pdfs/Net-BasedRealAnalysis.pdf) **1.9.10 **  # [Complex Analysis](https://dracentis.github.io/pdfs/ComplexAnalysis.pdf) **1.9.11 **  # [Abstract Algebra](https://dracentis.github.io/pdfs/AbstractAlgebra.pdf) **1.9.12 **  # [Fourier Analysis](https://dracentis.github.io/pdfs/FourierAnalysis.pdf) **1.9.13 **  # [Topology](https://dracentis.github.io/pdfs/Topology.pdf)