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\begin{document}

\begin{center}{\textbf{\large Quantum Condensed Matter Physics Cheatsheet} }\\
    \href{https://fc472.user.srcf.net/}{Feiyang}
\end{center}
\begin{multicols*}{3}

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\begin{topic}[Conduction]
    \textbf{Lorentz oscillator}
    \begin{align*}
        m \ddot{x} &=  - m \omega_T^2 x - m \gamma \dot{u}+ qE\\
        p &= \frac{N}{V} e x\\
        \chi &= \frac{P}{\epsilon_0 E}
    \end{align*}
    \textbf{Drude model}
    \begin{align*}
        \pqty{ \dv[]{t} + \frac{1}{\tau}} \bf j &=   \frac{N}{V} \frac{q^2}{m} \bf E + \frac{q}{m} (\bf j \crossproduct \bf B)\\
        \epsilon (\omega) &= \epsilon_\infty - \frac{\omega_P^2}{\omega^2 + i \omega /\tau}\qquad \omega_P^2 = \frac{ne^2}{m \epsilon_0}
    \end{align*}
\end{topic}

\begin{topic}[Hall effect]
    \textbf{Hall coefficient}
    \begin{align*}
        v_x &=  \frac{q \tau}{m} E_x\\
        B v_x &= E_y\\
        E_y &= \frac{qB \tau}{m} E_x \\
        j_x &= n v_x q\\
        R_H &= \frac{E_y}{j_x B} = \frac{B v_x}{j_x B} = \frac{1}{n q} 
    \end{align*}
\end{topic}



\begin{topic}
    [Thomas-Fermi screening]
    \begin{align*}
        q_{TF}^2 &= \frac{e^2 g_v(E_F)}{\epsilon_0}\\
        n(q) &=  \frac{\epsilon_0 q^2}{e} \frac{V_\text{ext} (q)}{1 + q^2 / q_{TF}^2}\\ 
        n (r) &= \frac{q_{TF}^2}{4\pi}  \frac{Q}{e} \frac{e^{- q_{TF} r}}{r}  
    \end{align*}
\end{topic}

\vfill\null
\columnbreak

\begin{topic}
    [Lattice heat capacity]
    \textbf{Debye model}
    \begin{align*}
        \omega &= v k\\
        D_D (\omega) &=  \frac{4\pi k^2}{(2\pi / L)^3} \dv[]{k}{\omega} = \frac{V \omega^2}{2\pi^2 v^3}\\
        N&= \int_0^{\omega_D} D_D(\omega) \dd{\omega} \\
        U &= \int \dd{\omega} D(\omega) \underbrace{n(\omega)}_\text{Bose-Einstein} \hbar \omega\\
        U &\propto \beta^{-4} \qquad\text{for } \beta \hbar\omega_D \gg 1\\
        C_V &= qN k_B \pqty{\frac{T}{\theta_D}}^3 \int_0^{\theta_D / T} \frac{ x^4 e^x \dd{x}}{(e^x -1)^2} 
    \end{align*}
    \textbf{electron gas}
    \begin{align*}
        u &= \int \dd{E} E g(E) \;\text{Fermi-Dirac}(E)\\
        c_V &\approx \frac{N}{V} \frac{k_B^2}{E_F} T
    \end{align*}
\end{topic}

\begin{topic}
    [Periodicity]
    \textbf{Bragg condition}: \( \bf q \dotproduct \bf R = 2\pi m \iff \bf q = \bf G\)\\
    \textbf{Bragg's law}: \(\bf k \dotproduct \bf G /2 = \pqty{G / 2}^2 \)\\
    \textbf{Periodic potential}: \(\bf \bqty{\hat{H}, \hat{T}} = 0\), \(\hat{T}\) is a symmetry\\
    \textbf{Bloch's theorem}
    \[
        \psi_q (x) = e^{iq r} \underbrace{u_{j,q} (r)}_\text{periodic}
    \]
\end{topic}

\columnbreak
\begin{topic}
    [Bloch state construction]
    \textbf{Nearly free electron model}
    \begin{align*}
        \ket{\psi_k} &= \sum c_{k - G} \ket{k- G}\\
        H &= \underbrace{T}_\text{diagonal} + \underbrace{V}_\text{periodic, off-diagonal}
    \end{align*}
    \textbf{Tight binding (LCAO)}\\
    one orbital per site, nearest neighbour only
    \begin{align*}
        H &= T+ V_a + V_b\\
        \pqty{T+ V_a} \ket{a}  &=  \ket{E_a}\\ 
        \pqty{T+ V_b} \ket{b}  &=  \ket{E_b}\\
        H &= \begin{pmatrix} E_a+ \bra{a}  V_b \ket{a} & t \\ t^* & E_b + \bra{b} V_a \ket{b}  \end{pmatrix} \\
        t &=  \bra{a}  \pqty{T+ V_a + V_b} \ket{b} \\
        E &= \bra{0} \hat{H} \ket{\psi} = \sum e^{i k R_n} \bra{0} \hat{H} \ket{n}
    \end{align*}
    \textbf{Pseudopotential}: smooth near the core, remove core (non-valence) \(e^-\).
\end{topic}

\begin{topic}
    [Bands]
    \textbf{Density of states of \(n\)-th band}\[
        g_n (E) = \int \frac{\dd{S}}{4\pi^3} \frac{1}{\abs{\grad_\perp E_n(k)}} 
    \]
    \textbf{holes}\begin{align*}
        k_h &= - k_e\\
        \epsilon_h (k_h) &=  - \epsilon_e (k_e)\\
        v_h &= v_e\\
        m_h^* &= - m_e^*
    \end{align*}
    \textbf{photoemission}
    \begin{align*}
        E_f &= E_i + \hbar \omega - \phi& k_{f \parallel} = k_{i \parallel}
    \end{align*}
\end{topic}
\newpage

\begin{topic}
    [Semiconductors]
    \textbf{Law of mass action} (intrinsic semiconductors)
    \begin{gather*}
        \abs{E - \mu} \gg k_B T\\
        n = \int_{E_c}^\infty \dd{E} g_e (E) f (E)\\
        p = \int_0^{E_v} \dd{E} g_h(E) f(E)\\
        np = n_c n_v e^{-\overbrace{E_g}^{E_c - E_v} \beta}
    \end{gather*}
    \textbf{doped} e.g. with donors of binding energy just below conduction band by \(\Delta_d\)
    \[
        n = \sqrt{n_c N_d} e^{-\Delta_d \beta /2}
    \]
    \textbf{currents}
    \begin{gather*}
        \begin{aligned}
            \text{generation (minority) } & - J^\text{gen} \\
            \text{recombination (majority) } &  J^\text{rec} = e^{- \beta e \pqty{\phi_b - V}} n_i^2 
        \end{aligned}\\
        I = n_i^2 e^{- \beta \overbrace{e\phi_b }^{E_g}} \pqty{e^{\beta eV} - 1}
    \end{gather*}
    A \emph{quantum well} traps carriers into bound states near a real space region to create lasers.
\end{topic}

\columnbreak
\begin{topic}
    [Magnetism]
    \textbf{Currie}
    \begin{align*}
        p(m) &= \frac{e^{- \beta E(m)}}{Z}\\
        m &= \int m p(m) \dd[2]{m}\\
        \chi &= \dv[]{m}{H} = \mu_0 \beta \left\langle m_i m_j\right\rangle \qquad \text{for } H = 0\\
        m^2 &= \frac{1}{3} \mu^2\\
        \chi &= \frac{1}{3} \frac{N}{V} \mu_0 my^2 \beta
    \end{align*}
    \textbf{Heisenberg}
    \begin{gather*}
        \text{(anti)clustering} \implies H\text{ depends on } \mathbf S \implies  \mathbf{S}_1 \dotproduct \mathbf S_2\\
        H_2 = - 2 J\; \mathbf{S}_1 \dotproduct \mathbf S_2 
    \end{gather*}
    \textbf{Pauli}

    Add in penalty \(U\) for doubly-filled states
    \begin{gather*}
        \begin{aligned}
            \epsilon_{k \uparrow} = \epsilon_k - \mu_B B_a + &U n_\downarrow\\
            \epsilon_{k \downarrow} = \epsilon_k + \mu_B B_a + &U n_\uparrow
        \end{aligned}\\
        \begin{aligned}
            \mu + \mu_B B_a &= E_{F \uparrow}\\
            \mu - \mu_B B_a &= E_{F \downarrow}\\
        \end{aligned}\\
        n_{\uparrow / \downarrow} = \int \dd{\epsilon} \frac{1}{2} g(\epsilon) f(\epsilon) = \int_0^{\mu_{\uparrow / \downarrow}} \dd{\epsilon}\frac{1}{2} g(\epsilon)\\
        \frac{N}{V} \pqty{n_\uparrow - n_\downarrow} = \frac{1}{2} g(E_F) \int_{\mu_\downarrow}^{\mu_\uparrow} \dd{\epsilon} \\
        M = \mu_B \pqty{n_\uparrow - n_\downarrow} \implies \chi_\sigma = \mu_0\frac{\mu_B^2 g(E_F)}{1- \frac{U g(E_F)}{2} }\\
        \frac{U g(E_F)}{2}  > 1 \implies \text{Ferromagnetism}
    \end{gather*}
\end{topic}

\columnbreak

\begin{topic}
    [Quantum oscillations]
    \begin{gather*}
        \oint p \dd{r} = \pqty{n+ \frac{1}{2}} h = q \oint \pqty{r \crossproduct B + A} \dd{r} = -q \Phi \\
        \Phi_n = A_r^{(n)} B_n = \pqty{n + \frac{1}{2}} \frac{h}{e}\\
        A_k = \pqty{\frac{Bq}{\hbar}}^2 A_r = \frac{2\pi e}{\hbar} B \pqty{n+ \frac{1}{2}} 
    \end{gather*}
\end{topic}

\begin{topic}
    [Fermi liquid]
    \[
        \Gamma \sim \epsilon^2
    \]
    \textbf{Quasiparticles}
    \begin{align*}
        \psi_k (\bf r, t) &= \psi_k (r) e^{-i \epsilon_k t / \hbar} e^{- \Gamma_k \abs{t}}\\
        A &= - \frac{1}{\pi} \Im\bqty{\frac{1}{\omega - \epsilon_k / \hbar + i \Gamma_k}}
    \end{align*}
    \textbf{Heavy Fermions} due to interaction
    \[
        g(E_F) \text{ high} \implies \text{magnet}
    \]
\end{topic}

\end{multicols*}
\end{document}