1.1 Recommended Textbooks
1.2 Presentation Topics
Definition 2.1.1 A quantum particle exhibits coherent motion iff its lifetime $t_\ell$ is much greater than $\frac{\hbar}{E}$.
\[t_\ell >> \frac{\hbar}{E}\]
Definition 2.1.2 A quantum particle exhibits incoherent motion iff its lifetime $t_\ell$ less than $\frac{\hbar}{E}$.
\[t_\ell \leq \frac{\hbar}{E}\]
Definition 2.1.3 A stationary 1D Brownian particle is a system of a stationary 1D particle experiencing many fast collisions with momentum $p_i$ integrated over a small time step $\Delta t$. Defined the force experienced by the particle $f(t)$ as a function of time.
\[f(t) = \frac{2}{\Delta t}\sum_i{p_i}\]
Definition 2.1.4 The ensemble average denoted $\langle x \rangle$ of a random variable $x$ is the integral of the variable weighted by the probability density function $P_{x}(x)$ across all possible values.
\[\langle x \rangle = \int_{-\infty}^{\infty}{x P_{x} dx}\]
Definition 2.1.5 The momentum diffusion coefficient is defined $D=2\nu\langle p_i^2\rangle = 2\nu m_{mol} k_B T$ where $\nu$ is the average frequency of collisions and $m^*$ is the effective mass of the particles.
Result 2.1.6 $\langle f\rangle = 0$
Result 2.1.7 $\langle f^2\rangle = \frac{4}{(\Delta t)^2} \sum_i{\langle p_i^2 \rangle}=\frac{4}{\Delta t}\nu\langle p_i^2\rangle = \frac{2D}{\Delta t}$
Result 2.1.8 $\langle f^{2n+1} \rangle = 0$
Result 2.1.9 $\langle f^{2n} \rangle = \langle f^2 \rangle^n (2n-1)!! =\left(\frac{4}{\Delta t}\nu\langle p_i^2\rangle\right)^n (2n-1)!! = \left(\frac{2D}{\Delta t}\right)^n (2n-1)!!$
Definition 2.1.10 A Gaussian Distribution with variance $\sigma=\langle f^2 \rangle$ is a distribution of the form:
\[P_f(x) = \frac{1}{\sqrt{2\pi\sigma}}e^{-x^2/2\sigma}\]
Definition 2.1.11 A probability density function of a variable is a function that when integrated over a region of possible values represents the probability of the variable being in that region.
Result 2.1.12 The probability density function of $f(t)$ at a particular time is Gaussian with $\sigma = \langle f^2 \rangle = \frac{2D}{\Delta t}$.
Definition 2.1.13 A functional probability density function is a probability density function that is integrated over different functions.
Result 2.1.14 The functional probability density of $f(t)$ is \[\mathcal{P}(f(t)) = C \text{exp}\left[{-\frac{1}{4D}\int_{-\infty}^{\infty}{f^2(t)dt}}\right]\]
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 $t>>\frac{1}{\Gamma}$, $\langle q^2\rangle = \langle (q(t)-q(0))^2 \rangle = 2\mathscr{D}\left[t - \frac{2}{2\Gamma}(1-e^{-2\Gamma t}) + \frac{1}{4\Gamma}(1-e^{-4\Gamma t})\right] \approx 2\mathscr{D}t$
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.
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
\[S_qq(\omega) = \frac{2\Gamma \omega_0^2}{(\omega^2-\omega_0^2)^2 + 4\Gamma^2\omega^2}\]
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
\[N(t) = N(0)e^{-W_{esc}t}\]
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
\[N(t) = N(0)e^{-W_{esc}t}\]
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 $\Psi_{\mathbf{k},n}$,
\[\mathbf{r} = i\partial_k + i\langle u_{\mathbf{k},n}(\mathbf{r}) | \nabla_{\mathbf{k}} u_{\mathbf{k},n}(\mathbf{r}) \rangle = i\partial_k + \mathcal{A}_n(\mathbf{r})\]\[\langle \mathbf{r} \rangle = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k}) \mathbf{r} c(\mathbf{k}) d\mathbf{k}} = \left( \frac{(2\pi)^d}{V} \right)^2\int{c^*(\mathbf{k})\left(i\partial_k + \mathcal{A}_{n,\mathbf{k}}(\mathbf{r})\right) c(\mathbf{k}) d\mathbf{k}}\]\[\langle \dot{\mathbf{r}} \rangle = \frac{\partial E_{\mathbf{k},n}}{\partial \mathbf{k}}\]
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
4.1 Impurity Scattering
4.2 Electric Conductivity
4.3 Thermal Conductivity
4.4 Magnetoconductivity
4.5 Cyclotron Resonance
5.1 Landauer 1D Conductivity
5.2 2D Electron Systems
5.3 Quantum Hall Effect
6.1 Weak Localization
6.2 Anderson Localization
6.3 Density Matrix and the Quantum Kinetic Equation
7.1 Electron-phonon Interaction
7.2 Polaronic Effect
7.3 Ohmic Dissipation
7.4 The Orthogonality Catastrophe
7.5 Holstein Polarons
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