Important Formulas in Quantum Mechanics

Time-Dependent Perturbation Theory

Solutions to Two-Energy Systems

\begin{equation} \dot c_a = -\frac{i}{\hbar}H_{ab}’ e^{-i\omega_0 t}c_b \end{equation} \begin{equation} \dot c_b = -\frac{i}{\hbar}H_{ba}’ e^{i\omega_0 t}c_a \end{equation}

1st Order Perturbation Approximation

\begin{equation} c_b^{(1)}(t) = -\frac{i}{\hbar}\int_0^t H_{ba}’(t’) e^{i\omega_0 t’}dt’ \end{equation}

Rate of Stimulated Emission

\begin{equation} R_{b\to a}(t) = \frac{\pi}{3\epsilon_0 \hbar^2} |\vec R|^2 \rho(\omega_0) \end{equation}

Einstein Constants

\begin{equation} A = \frac{\omega_0^3 \hbar}{\pi^2 c^3}B_{ab} = \frac{\omega_0^3 |\vec R|^2}{3\pi\epsilon_0\hbar c^3} \end{equation} \begin{equation} B_{ab}=B_{ba} = \frac{\pi |\vec R|^2}{3 \epsilon_0 \hbar^2} \end{equation}

Selection Rules

\begin{equation} \Delta m=\pm 1,0 \end{equation} \begin{equation} \Delta l = \pm 1 \end{equation}

Adiabatic Approximation

Adiabatic Theorem

\begin{equation} \Psi_n(t) = e^{i\theta_n(t)t}e^{i\gamma_n(t)} \psi_n(t) \end{equation}

Dynamic Phase

\begin{equation} \theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t’)dt’ \end{equation}

Geometric Phase

\begin{equation} \gamma_n(t) =i \int_0^t \langle \psi_n |\frac{\partial}{\partial R} \psi_n\rangle dR = i\int_{\vec R_i}^{\vec R_f} \langle \psi_n | \nabla_R\psi_n\rangle \cdot d\vec R \end{equation}

A-B Effect

\begin{equation} \gamma_n(T) = \frac{q}{\hbar} \int (\nabla \times \vec A)\cdot d\vec a = \frac{q\Phi}{\hbar} \end{equation}

Scattering

Radical Equation

\begin{equation} -\frac{\hbar^2}{2m}\frac{d^2 u}{dr^2}+[V(r) + \frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}]u = Eu \end{equation}

Sub-Wave Method

\begin{equation} \psi(r/\theta) = A\sum_{l=0}^{\infty} i^l (2l+1) [J_l(kr) + ika_l H_l^{(1)}(kr)]P_l(\cos \theta) \end{equation} \begin{equation} f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(\cos\theta) \end{equation}

1st Order Born Approximation

\begin{equation} f(\theta, \phi) = -\frac{m}{2\pi\hbar^2}\int e^{-i \Delta \vec k \cdot \vec r_0} V(\vec r_0) d^3 \vec r_0 \end{equation} \begin{equation} f(\theta, \phi) = -\frac{m}{2\pi\hbar^2}\int V(\vec r)d^3 \vec r \text{ (low energy)} \end{equation}

Landau Energy Level

Hamiltonian of 2DEG

\begin{equation} H = \frac{1}{2m}(\vec p -q \vec A)^2 - \mu\cdot \vec B = \hbar\omega(a^{\dagger} a +\frac{1}{2}) \end{equation}

Landau Gauge

\begin{equation} \vec A = (-yB,0,0) \end{equation}

Magnetic Length

\begin{equation} l_B = \sqrt{frac{\hbar}{eB}} \end{equation}

Ladder Operators

\begin{equation} a^{\dagger} = -\frac{1}{\sqrt{2}}[\frac{y-y_0}{l_B}-l_B \partial_y] \end{equation} \begin{equation} a = -\frac{1}{\sqrt{2}}[\frac{y-y_0}{l_B} + l_B \partial_y] \end{equation}

Eigen Energies of 2DEG

\begin{equation} \epsilon_n = \hbar\omega(n+\frac{1}{2}) \end{equation}

Landau Degeneracy

\begin{equation} N_L = \frac{1}{2\pi l_B^2} \end{equation}

2D Massless Dirac Model

\begin{equation} H = v(k_x\sigma_x +k_y\sigma_y) \end{equation}

Hamiltonian of 3DEG

\begin{equation} H = \frac{\hbar^2}{2m}(\vec k +\frac{e}{\hbar} \vec A)^2 \end{equation}

Landau Energies of 3DEG

\begin{equation} \epsilon_{k_z,n}=\hbar\omega(n+\frac{1}{2})+\frac{\hbar^2}{2m}k_z^2 \end{equation}

Landau Degeneracy

\begin{equation} N_L = \frac{1}{2\pi l_B^2} \end{equation}