Figure 1.2.15 Dulong Petit Law Figure
Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. By Nick B. - Own work, CC BY-SA 4.0, Link
These notes were written for the class PHY 801 at Michigan State University in Spring 2026 taught by Philip Crowley (orcid, google scholar).
Problem_set_1.pdf
Problem_set_2.pdf
Problem_set_3.pdf
Problem_set_4.pdf
Solutions_1.pdf
Solutions_2.pdf
Part 1 - Heat capacity.pdf
Part 1 - Slides.pdf
Part 2 - Drude-Lorentz theory.pdf
Part 2 - Supplementary Note.pdf
Part 2 - Slides.pdf
Part 3 - Pauli Exclusion and Sommerfeld.pdf
Part 3 - Supplementary Note.pdf
Part 3 - Slides.pdf
Part 4 - Hartree Fock and Screening.pdf
These notes are very closely related to Statistical Mechanics. However, there are some notable notation differences between the two texts. Some of these differences include: total energy $U$ instead of $E$, multiplicity function $W$ instead of $\Omega$ and many others. The texts are self consistent, but cross comparison should be done carefully.
Definition 1.1.1 Energy denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by work and heat.
Definition 1.1.2 Work is energy transferred to a system by macroscopic forces.
Definition 1.1.3 Heat is energy transferred to a system by microscopic forces.
Definition 1.1.4 The multiplicity function $W$ of a system is the number of possible microstates for a given macrostate.
Definition 1.1.5 The Boltzmann constant denoted $k_B$ is the the proportionality factor fixed at exactly $k_B=1.380649\times 10^{-23}\text{J}/\text{K}$ that defines temperature as it is related to the statistical probability of energy states in a system.
Definition 1.1.6 The Plank constant denoted $h$ is the proportionality factor fixed at exactly $h=6.62607015\times10^{-34}\text{J}/\text{Hz}$ relating a photon's energy to it's frequency.
Definition 1.1.7 The reduced Plank constant denoted $\hbar=h/2\pi$ where $h$ is the Plank constant.
Definition 1.1.8 The enthalpy is a state function $H$ defined for total energy $U$, pressures $\mathbf{P}$ and volumes $\mathbf{V}$.
\[H = U+\mathbf{P}\cdot\mathbf{V}\]
Definition 1.1.9 The Helmholtz free energy is a state function $F$ defined for total energy $U$, pressures $\mathbf{P}$ and volumes $\mathbf{V}$.
\[F = U-TS\]
Definition 1.1.10 The entropy $S$ of a system is the Boltzmann constant times the natural log of the multiplicity function.
\[S = k_B\log W\]
Definition 1.1.11 The temperature $T$ in units of kelvin (K) and thermodynamic temperature $\beta$ in units of joules (J) of a system are defined in terms of the derivative of energy $U$ with respect to entropy $S$.
\[T = \frac{\partial U}{\partial S} = \frac{1}{k_B\beta},\quad\beta = \frac{1}{k_B}\frac{\partial S}{\partial U} = \frac{1}{k_B T}\]
Definition 1.1.12 The microcanonical ensemble is the ensemble of statistical mechanics where the macrostates are described by the total energy $U$, volumes $\mathbf{V}$ and particle numbers $\mathbf{N}$. The probability of each possible microstate $\mathscr{p}_i$ is assumed to be the same, so it is simply one over the multiplicity function $W$, which can be written exactly as the number of states that match the macrostate. The canonical ensemble and the grand canonical ensemble can be derived by considering a system inside a large reservoir in the microcanonical ensemble.
\[\mathscr{p}_i = \frac{1}{W(U,\mathbf{V},\mathbf{N})}\]
Definition 1.1.13 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 $z$.
\[\mathscr{p}_i = \frac{1}{W(T,\mathbf{V},\mathbf{N})}=\frac{e^{-\beta E_i}}{\sum_{j}{e^{-\beta E_j}}} = \frac{e^{-\beta E_i}}{z}\]\[z = \sum_{j}{e^{-\beta E_j}} = \sum_{j}{e^{-E_j/(k_BT)}}\]
Definition 1.1.14 The canonical total energy U of a system in the canonical ensemble is the ensemble average of the total energy of the system.
\[U = \sum_{i}{E_i \mathscr{p}_i} = \frac{1}{z}\sum_{i}{\frac{-\partial}{\partial \beta}e^{-\beta E_i}} = -\frac{1}{z}\frac{\partial z}{\partial \beta} = -\frac{\partial}{\partial \beta}\log z\]
Definition 1.1.15 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 $\mathscr{z}$.
\[\mathscr{p}_i = \frac{1}{W(\mathbf{T},\mathbf{V},\mathbf{\mu})}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}}=\frac{e^{-\beta(E_i-\mathbf{\mu}\cdot\mathbf{N}_i)}}{\mathscr{z}}\]\[\mathscr{z} = \sum_\mathbf{N}{\sum_j{e^{-\beta(E_j-\mathbf{\mu}\cdot\mathbf{N})}}}\]
Definition 1.2.1 Energy denoted $U$ with SI unit Joules (J) is a conserved quantity that is transferred between systems by work and heat.
Definition 1.2.2 The heat capacity denoted $C$ is the derivative of total energy $U$ in terms of temperature $T$ of a system.
\[C = \frac{\partial U}{\partial T}\]
Definition 1.2.3 The heat capacity at constant pressure denoted $C_P$ is the derivative of total enthalpy $H$ in terms of temperature $T$ of a system while pressure $P$ is held constant.
\[C_P = \left(\frac{\partial H}{\partial T}\right)_P\]
Definition 1.2.4 The heat capacity at constant volume denoted $C_V$ is the derivative of total energy $U$ in terms of temperature $T$ of a system while pressure $V$ is held constant.
\[C_V = \left(\frac{\partial U}{\partial T}\right)_V\]
Definition 1.2.5 The coefficient of thermal expansion denoted $\alpha$ is the derivative of volume $V$ in terms of temperature $T$ of a system while pressure $P$ is held constant divided by the volume of the system.
\[\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P>0\]
Definition 1.2.6 The isothermal compressibility denoted $\kappa$ is the negative derivative of volume $V$ in terms of pressure $P$ of a system while temperature $T$ is held constant divided by the volume of the system.
\[\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T>0\]
Result 1.2.7 Mayer's relation states that the heat capacity at constant pressure $C_P$ and heat capacity at constant volume $C_V$ must differ by a positive nonzero value determined by the temperature $T$, volume $V$, coefficient of thermal expansion $\alpha$ and isothermal compressibility $\kappa$ of the system.
\[C_P-C_V = \frac{TV\alpha^2}{\kappa}= -T\left(\frac{\partial V}{\partial T}\right)^2_P\left(\frac{\partial P}{\partial V}\right)_T>0\]
Definition 1.2.8 An intensive quantity is a variable of a system that does not scale with the size of the system.,
Definition 1.2.9 An extensive quantity is a variable of a system that does scale with the size of the system.
Definition 1.2.10 The specific heat denoted $c$ is the derivative of total energy $U$ in terms of temperature $T$ scaled per unit mass $M$ of the system such that it is an intensive quantity.
\[c = \frac{C}{M} = \frac{1}{M}\frac{\partial U}{\partial T}\]
Definition 1.2.11 The molar specific heat denoted $c_{mol}$ is the derivative of total energy $U$ in terms of temperature $T$ scaled to the number of moles $N_{mol}$ in the system such that it is an intensive quantity. It can also be defined in terms of the specific heat $c$ and molar mass $m_{mol}$.
\[c_{mol} = \frac{C}{N_{mol}} = \frac{c}{m_{mol}} = \frac{1}{N_{mol}}\frac{\partial U}{\partial T}\]
Definition 1.2.12 The Boltzmann solid is a classical model of solids in the canonical ensemble that models the valence electrons of atoms as classical particles in potential wells with the following Hamiltonian $H$, where $\vec{p}$ is the momentum, $m$ is the mass, $k$ is the spring constant and $\vec{x}$ is position of the electron relative to the center of the potential well.
\[H = \frac{p^2}{2m}+\frac{1}{2}kx^2\]\[\mathscr{p}(\vec{x},\vec{p}) = \frac{e^{-\beta H(\vec{x},\vec{p})}}{z},\quad z = \int d^3\vec{x}\int d^3\vec{p} e^{-\beta H(\vec{x},\vec{p})}\]
Result 1.2.13 The intensive energy of a Boltzmann solid $u$ is the average energy per particle in a 3d solid determined by the following relation with temperature $T$.
\[u = -\frac{\partial}{\partial \beta}\log z = 3k_BT\]
Law 1.2.14 The Dulong Petit law states that the molar specific heat for most bulk materials is a constant at high temperatures.
\[C = N_{atoms}\frac{\partial u}{\partial T} = 3k_BN_{atoms}\]\[c_{\text{mol}}=\frac{C}{N_{mol}}=\frac{3k_BN_{atoms}}{N_{mol}} = 3k_BN_{Avogadro} = 3R\]
Figure 1.2.15 Dulong Petit Law Figure
Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. By Nick B. - Own work, CC BY-SA 4.0, Link
Definition 1.3.1 The Einstein solid is a quantum model of solids in the canonical ensemble that models the valence electrons of atoms as quantum harmonic oscillators with the following Hamiltonian $H$, where $\hat{p}$ is the momentum operator, $m$ is the mass, $k$ is the spring constant and $\hat{x}$ is the position operator.
\[\hat{H} = \frac{\hat{p}^2}{2m}+\frac{1}{2}k\hat{x}^2\]
Result 1.3.2 The eigenvalues of the 1D Harmonic oscillator $E_n$ for the corresponding eigenstates $\ket{n}$ are given by the following equation for non-negative integer $n\in\mathbb{N}$.
\[E_n = \hbar\omega\left(n+\frac{1}{2}\right),\quad H\ket{n} = E_n\ket{n}, \quad \omega = \sqrt\frac{k}{m}\]
Proposition 1.3.3 Geometric series convergence states that for $|r|<1$ the following infinite series converges to $1/(1-r)$.
\[\sum_{k=0}^\infty{r^k} = \frac{1}{1-r}\]
Result 1.3.4 The partition function of a 1D harmonic oscillator $z_{1D}$ can be written as a geometric series:
\[z_{1D} = \sum_{n=0}^\infty{e^{-\beta E_n}} = \sum_{n=0}^\infty{e^{-\beta\hbar\omega(n+\frac{1}{2})}} = \frac{e^{\beta\hbar\omega/2}}{e^{\beta\hbar\omega}-1}\]
Result 1.3.5 The eigenvalues of the 3D Harmonic oscillator $E_{n_x,n_y,n_z}$ for the corresponding eigenstates $\ket{n_x,n_y,n_z}$ are given by the following equation for non-negative integers $n_x,n_y,n_z\in\mathbb{N}$.
\[E_{n_x,n_y,n_z} = \hbar\omega\left(n_x+n_y+n_z+\frac{3}{2}\right)\]\[H\ket{n_z,n_y,n_z} = E_{n_z,n_y,n_z}\ket{n}\]\[\omega = \sqrt\frac{k}{m}\]
Result 1.3.6 The partition function of a 3D harmonic oscillator $z_{3D}$ can be written in terms of the partition function for the 1D harmonic oscillator $z_{1D}$:
\[z_{3D} = \sum_{n_x,n_y,n_z=0}^\infty{e^{-\beta E_{n_x,n_y,n_z}}} = z_{1D}^3 = \left(\frac{e^{\beta\hbar\omega/2}}{e^{\beta\hbar\omega}-1}\right)^3\]
Definition 1.3.7 The Bose factor is $n_B(\beta\hbar\omega) = \frac{1}{e^{\beta\hbar\omega}-1}$.
Result 1.3.8 The intensive energy of a Einstein solid $u$ is the average energy per particle in a 3d solid determined by the following relation with temperature $T$.
\[u = -\frac{\partial}{\partial \beta}\log z_{3D} = -3\frac{\partial}{\partial \beta}\log z_{1D} \]\[= 3\hbar\omega \left(n_B(\beta\hbar\omega)+\frac{1}{2} \right) = 3\hbar\omega \left( \frac{1}{e^{\beta\hbar\omega}-1} + \frac{1}{2} \right)\]
Result 1.3.9 The molar heat capacity of an Einstain solid $c_{\text{mol}}$ satisfies the Dulong Petit law at high temperatures as $T\to\infty$ while converging to zero as $T\to0$.
\[c_{\text{mo}l} = \frac{C}{N_{mol}} = N_{Avogadro}\frac{\partial u}{\partial T} = 3R(\beta\hbar\omega)^2\frac{e^{\beta\hbar\omega}}{\left( e^{\beta\hbar\omega}-1 \right)^2}\]
Definition 1.3.10 The einstein temperature denoted $T_E$ is the critical temperature where the molar heat capacity of an Einstein solid starts decreasing.
\[T_E = \frac{\hbar\omega}{k_B}\]
Figure 1.3.11 Molar Heat Capacity of an Einstein Solid vs Temperature
Molar heat capacity predicted for an Einstein solid as a function of temperature. Public Domain, own work.
Also see the Wikipedia article for this model: https://en.wikipedia.org/wiki/Debye_model
Definition 1.4.1 The Debye solid is a model of solids in the canonical ensemble that models the collective phononic collective modes the atoms in the solid for some speed of sound $v$, the frequencies of the collect modes $\omega_k$ for wave number $\vec{k}$ are modeled by the following equations for the total energy $U$.
\[\omega_\vec{k} = v\abs{\vec{k}}\]\[U = 3\sum_\vec{k}\hbar\omega_\vec{k}\left( n_B(\beta\hbar\omega_\vec{k}) + \frac{1}{2} \right) = 3\sum_\vec{k}\hbar\omega_\vec{k}\left( \frac{1}{e^{\beta\hbar\omega_\vec{k}}-1} + \frac{1}{2} \right)\]
Definition 1.4.2 To calculate the modes of a Debye solid we assume periodic boundary conditions for some distance $L$ which is very large compared to the scale of the atom as to include all the lower frequency modes.
\[\vec{k}L = 2\pi\vec{n}\]
Definition 1.4.3 The Debye frequency denoted $\omega_D$ is the maximum frequency of phonons in a Debye solid, defined in terms of the density $\rho$ and speed of sound $v$ of the solid.
\[\omega_D = \left(6\pi^2\rho\right)^{1/3}v\]
Definition 1.4.4 The Debye temperature is $T_D = \frac{\hbar\omega_D}{k_B}$ where $\omega_D$ is the debye frequency.
Definition 1.4.5 The Debye density of states denoted $g(\omega)$ of frequency modes $\omega$ with periodic boundary conditions $L$ and volume $V$ is the following function.
\[g(\omega) = \frac{L^3\omega^2}{2\pi^2V^3} = \frac{3N\omega^2}{\omega_D^3}\]
Result 1.4.6 The sum of an isotropic function $f(\omega_{k})$ for all wave numbers $\vec{k}$ can be approximated as an integral of the density of states $g(\omega_{\vec{k}})$ and the function $f(\omega_{\vec{k}})$.
\[\sum_{\vec{k}}f(\omega_\vec{k})=\sum_\vec{k}f(v\abs{\vec{k}})=\sum_{\vec{n}\in\mathbb{N}^3}f\left(\frac{2\pi v}{L}\abs{\vec{n}}\right)\]\[\approx\int d^3\vec{n} f\left(\frac{2\pi v}{L}\abs{\vec{n}}\right) = \left(\frac{L}{2\pi}\right)^3\int d^3\vec{k}f\left(v\abs{\vec{k}}\right)\]\[= \frac{L^3}{2\pi^2}\int_0^{\omega_D} dk k^2 f(vk) = \frac{L^3}{2\pi^2 v^3}\int_0^{\omega_D} d\omega \omega^2 f(\omega) = \int_0^{\omega_D} d\omega g(\omega) f(\omega)\]
We set the maximum frequency of the integrate to $\omega_D$ because there are a finite number of atoms $N$. It turns out that there is a maximum frequency $\omega$ that these collective modes can exhibit. The next result proves that this cutoff frequency is indeed the Debye frequency $\omega_D$.
Result 1.4.8 The Debye frequency is the maximum frequency in a Debye solid, because there are a finite number of atoms $N$ in a solid.
\[N = \sum_k 1 = \int_0^{\omega_D} d\omega g(\omega) = \int_0^{\omega_D} d\omega \frac{3N\omega^2}{\omega_D^3} = N\frac{\omega_D^3}{\omega_D^3} = N\]
Result 1.4.9 The total energy $U$ of a Debye solid is given by the following integral of the Debye density of states $g(\omega)$ and the Bose factor $n_B(\beta\hbar\omega)$.
\[U = \int_0^{\omega_D} d\omega g(\omega) 3\hbar\omega\left( n_B(\beta\hbar\omega) + \frac{1}{2} \right) = \int_0^{\omega_D} d\omega \frac{3N\omega^2}{\omega_D^3} 3\hbar\omega\left( \frac{1}{e^{\beta\hbar\omega}-1} + \frac{1}{2} \right)\]
Result 1.4.10 The molar heat capacity $c_{mol}$ of a Debye solid is given by the following integral.
\[c_{mol} = \frac{1}{N_{mol}}\frac{\partial U}{\partial T} = \frac{1}{N_{mol}}\frac{\partial}{\partial T}\int_0^{\omega_D} d\omega g(\omega) 3\hbar\omega\left( n_B(\beta\hbar\omega) + \frac{1}{2} \right) \]\[= \frac{1}{N_{mol}}\frac{\partial}{\partial T}\int_0^{\omega_D} d\omega \frac{3N\omega^2}{\omega_D^3} 3\hbar\omega\left( \frac{1}{e^{\beta\hbar\omega}-1} + \frac{1}{2} \right)\]\[=3k_BN_{Avogadro}\frac{(\beta\hbar\omega_D)^2e^{\beta\hbar\omega} }{(e^{\beta\hbar\omega}-1)^2} =3R\frac{(\beta\hbar\omega_D)^2e^{\beta\hbar\omega} }{(e^{\beta\hbar\omega}-1)^2}\]
Figure 1.4.11 Debye vs. Einstein
Predicted heat capacity as a function of temperature. Public Domain, Link
Definition 2.1.1 The Drude model is a simple model of electron motion and scattering in a material with a differential equation describing of the momentum $\vec{p}$ of electrons experiencing the Lorentz force from an external electric field $\vec{E}$, magnetic field $\vec{B}$ and scattering with a mean scattering time of $\tau$.
\[\frac{\partial \vec{p}}{\partial t} = -e\left( \vec{E} + \frac{1}{m}\vec{p}\times\vec{B} \right) + \frac{\vec{p}}{\tau}\]
Despite the apparent simplicity of the Drude Model, it has been wildly successful at predicting a variety of phenomena related to electron transport in solids. Some of these phenomena include:
The following sections will describe each of these phenomena using the Drude Model. The drude model can then also be expanded into a complete thermodynamic theory for electrons with the Drude-Lorentz gas. Which allows it to at least conceptually explain the following phenomena:
Definition 2.2.1 A current density denoted $\vec{j}$ is a vector field describing the average density of charge flowing through a particular point in space per second.
Definition 2.2.2 The electric conductivity denoted $\sigma$ of a material is the coefficient or tensor that relates the electric field $\vec{E}$ to the current density $\vec{j}$.
\[\vec{j} = \sigma \vec{E}\]
Definition 2.2.3 The resistivity denoted $\rho$ of a material is the coefficient or tensor that relates the current density $\vec{\rho}$ flowing through a material with the electric field $\vec{E}$ required to drive that current.
\[\vec{E} = \rho\vec{j}\]
Corollary 2.2.4 The conductivity $\sigma$ and resistivity $\rho$ of a material are inverses of each other.
\[\sigma = \frac{1}{\rho},\quad \rho = \frac{1}{\sigma}\]
Definition 2.2.5 The Drude conducitivity denoted $\sigma_D$ is the electric conductivity predicted by the Drude model for a pure electric field $\vec{E}$ ($\vec{B}=\vec{0}$) where $\tau$ is the mean scattering time, $n_e$ is number of electrons, $e$ is the elemental charge and $m_e$ is the mass of charge carriers.
\[\vec{j}_D = \frac{e^2n_e\tau}{m_e}\vec{E} = \sigma_D\vec{E}\]
Definition 2.2.6 The resistance denoted $R$ of a prism of material with cross sectional area $A$, length $L$ and resistivity $\rho$ is given by the following relation.
\[R = \rho \frac{\ell}{A}\]
Law 2.2.7 Ohm's law states that the total current $I$ flowing through a material is equal to the resistance $R$ times the bias voltage across the material $V$.
\[V = IR\]
Definition 2.3.1 The cyclotron frequency denoted $\omega_c$ is the frequency at which an electron would spin in a magnetic field of strength $B$, with elemental charge $e$ and electron mass $m_e$.
\[\omega_c = \frac{eB}{m_e}\]
Definition 2.3.2 The hall effect is the production of an electric field $\vec{E}_{\text{hall}}$ (called the hall field) across a material in the direction of the cross product between the external electric field $\vec{E}_{\text{ext}}$ and the magnetic field $\vec{B}$.
Result 2.3.3 The classical hall effect is the hall effect as predicted by solving the equilibrium condition $\frac{\partial \vec{p}}{\partial t} = 0$ for the Drude Model in 2 dimensions with a magnetic field $\vec{B}=B\hat{z}$ perpendicular to the plane and an in-plane electric field $\vec{E} = E_x\hat{x} + E_y\hat{y}$.
\[0=-eE_x - \omega_c p_y - \frac{p_x}{\tau}, \quad 0=-eE_y + \omega_c p_x - \frac{p_y}{\tau}\]\[\vec{p} = \frac{-e\tau}{1+(\omega_c\tau)^2}\begin{pmatrix}1 & -\omega_c\tau\\ \omega_c\tau & 1\end{pmatrix}\vec{E}\]
Figure 2.3.4 Hall Effect Diagram
Hall Effect Measurement Setup for Electrons. An external field $E_x$ is applied in the x direction and a magnetic field $B_z$ is applied in the z direction, resulting in a hall field $E_y$ in the y direction. Public Domain, Link
Result 2.3.5 The classical hall field $\vec{E}_{\text{hall}}$ for external electric field $\vec{E}_{\text{ext}}$ and magnetic field $\vec{B}$ can be written in terms of the cyclotron frequency $\omega_c$ and mean scattering time $\tau$
\[\vec{E}_{\text{hall}} = \omega_c\tau(\vec{B}\times\vec{E}_{\text{ext}})\]
Definition 2.3.6 The hall coefficient denoted $R_H$ is the measurable ratio between the hall field $E_y$ and the product of the current $J_x$ and $B_z$ applied to drive that hall field. The Drude model predicts that this quantity is related to the charge carrier density $n_e$.
\[R_H = \frac{E_y}{J_xB_z} = \frac{-\omega_c\tau E_x}{J_x B_z} = \frac{-1}{en_e}\]
Definition 2.4.1 The temperature gradient denoted $\vec{\nabla} T(\vec{r})$ is the the gradient of temperature $T(\vec{r})$ at position $\vec{r}$ in a material.
Definition 2.4.2 The heat current denoted $\vec{j}_q$ is the rate of energy transfer through a material due to temperature gradient.
Definition 2.4.3 The thermal conductivity denoted $\kappa$ of a material is the coefficient that relates the temperature gradient $\vec\nabla T$ to the heat current $\vec{j}_q$.
\[\vec{j}_q = -\kappa\vec\nabla T\]
Definition 2.4.4 The Drude thermal conductivity denoted $\kappa_D$ is the thermal conductivity predicted by the predicted by the Drude model with density of electron $n$, average thermal velocity $v = \frac{k_B T}{m_e}$, mean scattering time $\tau$ and molar heat capacity $c_{\text{el,mol}} = \frac{\partial u}{\partial T}$
\[\kappa_D = nv^2\tau\frac{\partial u}{\partial T} = n \frac{k_BT}{m_e}\tau c_{\text{el,mol}}\]
Definition 2.4.5 The Lorentz number denoted $L$ is the proportionality constant that relates the thermal conductivity $\kappa$ to the electric conductivity $\sigma$ at temperature $T$.
\[L = \frac{\kappa}{\sigma T}\]
Result 2.4.6 The Drude Weiderman Franz Law describes the Lorentz Number predicted by the Drude model.
\[L = \frac{\kappa}{\sigma T} = \frac{n \frac{k_BT}{m_e}\tau c_{\text{el,mol}}}{\frac{e^2n_e\tau}{m_e} T} = \frac{k_B c_{\text{el,mol}}}{e^2} = \frac{3}{2}\left(\frac{k_B}{e}\right)^2\]
Definition 2.5.1 The distribution function denoted $f(\vec{r},\vec{p})$ for classical systems is the density of particle in position-momentum phase space. For $N$ particles at positions $\vec{r}_n$ and momenta $\vec{p}_n$, is given by the following sum of 3D Dirac deltas $\delta^3$.
\[f(\vec{r},\vec{p}) = \sum_{n=1}^N\delta^3(\vec{r}-\vec{r}_n)\delta^3(\vec{p}-\vec{p}_n)\]
Definition 2.5.2 A generic physical observable $A$ of distribution function $f(\vec{r},\vec{p})$ is given the following integral where $A(\vec{r},\vec{p})$ is the contribution to that observable by a single particle in the distribution function.
\[A = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})A(\vec{r},\vec(p))\]
Result 2.5.3 The particle number $N$ of a distribution function $f(\vec{r},\vec{p})$ is given by $N = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})$.
Result 2.5.4 The particle density $n(\vec{r})$ of a distribution function $f(\vec{r},\vec{p})$ is given by $n(\vec{r}) = \int d^3\vec{p} f(\vec{r},\vec{p})$.
Result 2.5.5 The current density $\vec{j}(\vec{r})$ of a distribution function $f(\vec{r},\vec{p})$ is given by $\vec{j}(\vec{r}) = \int d^3\vec{p} f(\vec{r},\vec{p}) \left(\frac{-e\vec{p}}{m}\right)$.
Result 2.5.6 The total energy $U$ of a distribution function $f(\vec{r},\vec{p})$ is given by $U = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p})$.
Result 2.5.7 The total energy $\vec{p}$ of a distribution function $f(\vec{r},\vec{p})$ is given by $\vec{p} = \int d^3\vec{r}d^3\vec{p} f(\vec{r},\vec{p}) \vec{p}$.
Definition 2.5.8 A Drude-Lorentz gas is a classical model of electrons in a solid that assume electrons relax to a Maxwell-Boltzmann equilibrium via simple random scattering with mean lifetime $\tau$. See the supplementary note for a guided derivation. The model describes the behavior of a distribution $f(\vec{r},\vec{p})$ of electrons with the following differential equation such that the equilibrium condition is the Maxwell-Boltzmann distribution $f_0(\vec{r},\vec{p})$.
\[\frac{d}{dt}f(\vec{r},\vec{p}) = \frac{f_0(\vec{r},\vec{p})-f(\vec{r},\vec{p})}{\tau}-\frac{\partial \vec{r}}{\partial t}\cdot\nabla_{\vec{r}}f(\vec{r},\vec{p}) - \frac{\partial \vec{p}}{\partial t}\cdot\nabla_{\vec{p}}f(\vec{r},\vec{p})\]\[f_0(\vec{r},\vec{p}) = \frac{n e^{-\beta p^2/(2m)}}{(2\pi m_e k_B T)^{3/2}}\]
Result 2.5.9 For weak fields $\vec{E}$ and $\vec\nabla T$, the distribution of the Drude-Lortenz Gas can be approximated by the following.
\[f(\vec{r},\vec{p}) \approx f_0 + \frac{\partial f}{\partial E}\vec{E} + \frac{\partial f}{\partial \vec\nabla T} \vec{\nabla}T\]
Definition 2.5.10 The linear response operators denoted $L_{\alpha\beta}$ for $\alpha,\beta\in\{1,2\}$ describe the linear relationship for small electric fields $\vec{E}$ and temperature gradients $\vec\nabla T$ with the current $\vec{j}$ and heat current $\vec{j}_q$.
\[\begin{pmatrix} \vec{j}\\ \vec{j}_q \end{pmatrix} = \begin{pmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{pmatrix}\begin{pmatrix} \vec{E} \\ -\vec{\nabla}T\end{pmatrix}\]
Result 2.5.11 The Drude-Lorentz electric conductivity is the electric conductivity $\sigma$ predicted by the Drude-Lorentz model for small electric fields $\vec{E}$ and $\vec\nabla T =0$,
\[\vec{j} = \sigma \vec{E} = L_{11} \vec{E} = \frac{ne\tau}{m_e}\vec{E}\]
Result 2.5.12 The Drude-Lorentz heat conductivity is the heat conductivity $\kappa$ predicted bt the Drude-Lorentz model for small temperature gradients $\vec\nabla T$ and $\vec{E} = 0$,
\[\vec{j}_q = -\kappa \vec\nabla T = -\left(L_{22} - \frac{L_{12}L_{21}}{L_{22}}\right)\vec\nabla T = -\frac{5}{2}\frac{k_B^2 nT\tau}{m_e}\vec\nabla T\]
Result 2.5.13 The Drude-Lorentz seebeck coefficient is the seebeck coefficient $S$ predicted by the Drude-Lorentz model for small temperature gradients $\vec\nabla T$ and $\vec{j} = 0$,
\[\vec{E} = S\vec\nabla T = \frac{L_{12}}{L_{11}}\vec\nabla T = -\frac{k_B}{e} \vec\nabla T\]
Result 2.5.14 The Drude-Lorentz peltiercoefficient is the peltier coefficient $\Pi$ predicted by the Drude-Lorentz model for small currents $\vec{j}\neq 0$ and $\vec\nabla T = 0$,
\[\vec{j}_q = \Pi \vec{j} = ST\vec{j} = \frac{k_BT}{e}\vec{j}\]
Result 2.5.15 The Drude-Lorentz molar heat capacity is the molar heat capacity $c_{\text{el,mol}}$ predicted by the Drude-Lorentz model,
\[c_{\text{el,mol}} = \frac{3}{2} R Z\]
Law 2.5.16 The Drude-Lorentz Wiedemann–Franz law describes the Lorentz Number predicted by the Drude-Lorentz model.
\[L = \frac{\kappa}{\sigma L} = \frac{5}{2}\left( \frac{k_B}{e} \right)^2\]
Note 2.5.17 While the Drude-Lorentz gas successfully predicts the existence of these effects, they value of the coefficients are off by orders of magnitude.
Definition 2.6.1 A plane wave with frequency $\omega$ is an electric field $\vec{E}(\vec{r},t)$ of the following form.
\[\vec{E}(\vec{r},t) = \vec{E}_0\cos(\vec{k}\cdot\vec{r} - \omega t)\]
Definition 2.6.2 The Drude plasma frequency denoted $\omega_p$ is the plasma frequency of a Drude metal.
Result 2.6.3 For $\omega\tau >> 1$
\[\omega_p = \sqrt\frac{\sigma_0}{\varepsilon_0 \tau} = \sqrt{\frac{ne^2}{m_e\varepsilon_0}}\]
Result 2.6.4 Penetration depth
Result 2.6.5 wavelengths at high frequency
Result 2.6.6 plasma wavelength
Definition 3.1.1 The single electron Hamiltonian is the quantum mechanical Hamiltonian consisting on a single electron.
\[H = \frac{p^2}{2m},\quad \vec{p} = -i\hbar\vec\nabla_r\]
Result 3.1.2 The energy eigenstates of a single electron with volume $V=L^3$ are given by the following equation for positive integers $n_x,n_y,n_z\in\mathbb{N}$ and spins $\sigma \in \{1,-1\}$,
\[\phi_{\vec{k},\sigma}(\vec{r}) = \frac{1}{\sqrt{V}}e^{i\vec{k}\cdot\vec{r}}\]\[E(\vec{k},\sigma) = \frac{\hbar^2 k^2}{2m},\quad \vec{k} = \frac{2\pi}{L}\vec{n}\]
Definition 3.2.1 The two electron Hamiltonian is the quantum mechanical Hamiltonian consisting of two electrons with spin.
\[H = \sum_{n=1,2}\frac{p_n^2}{2m},\quad \vec{p}_n = -i\hbar\vec\nabla_r\]
The two electron system must also have exchange symmetry, that is energy states are indistinguishable under particle exchange.
Definition 3.2.2 The exchange operator denoted $\hat{P}$ is the operator that swaps the particles of a state.
\[\hat{P}\Psi(\vec{r}_1,\sigma_1,\vec{r}_2,\sigma_2) = \Psi(\vec{r}_2,\sigma_2,\vec{r}_1,\sigma_1)\]\[\hat{P}^2\Psi = \Psi\]
Definition 3.2.3 A boson is a particle whose bare states are unaffected by particle exchange,
\[\hat{P}\Psi = \Psi\]
Definition 3.2.4 A fermion is a particle whose bare states are negated by particle exchange,
\[\hat{P}\Psi = -\Psi\]
Result 3.2.5 No bare states, states must not change under particle exchange.
\[\Psi = \frac{1}{\sqrt{2}}\text{det}\left[\right]^2\]
Section 3.3
Section 3.4
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 5.1
Section 5.2
Section 5.3
Section 5.4
Section 6.1
Section 6.2
Section 6.3
Section 6.4
Section 7.1
Section 7.2
Section 7.3
Section 8.1
Section 8.2
Section 8.3
