Equation
Published:
Some random equations
Classical Mechanics / 古典力学
\[\begin{aligned} & S = \int_{t_1}^{t_2} L\left(q_i,\dot q_i,t\right)\,dt \\ & \delta S = 0 \end{aligned}\]
Electromagnetism / 電磁気学
\[\begin{aligned} &\nabla\cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\ &\nabla\times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla\cdot \mathbf{B} = 0 \\ &\nabla\times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}\]
Statistical Mechanics / 統計力学
\[\begin{aligned} & S = - k_B \sum_i p_i \ln p_i \\ & \delta S = 0 \\ & dU = T\,dS - P\,dV + \mu\,dN \end{aligned}\]
Quantum Mechanics / 量子力学
\[\begin{aligned} & |\psi\rangle \in \mathcal{H} \\ & \hat{A} = \hat{A}^\dagger \\ & P(a_n) = |\langle a_n | \psi \rangle|^2 \\ & i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H}|\psi(t)\rangle \end{aligned}\]
Formulation / 定式化
State / 状態
\[\begin{aligned} & \langle m | n \rangle = \delta_{mn} && \langle a | a' \rangle = \delta(a - a') \\ & \sum_n |n\rangle\langle n| = \mathbb I && \int |a\rangle\langle a| \, da = \mathbb I \\ & |\psi\rangle = \sum_n |n\rangle\langle n|\psi\rangle && |\psi\rangle = \int |a\rangle\langle a|\psi\rangle \, da \end{aligned}\]
Observable / オブザーバブル
\[\begin{aligned} & \hat{A}|a\rangle = a|a\rangle && \langle a | a \rangle = \mathbb I \\ & a = a^* && \langle a_n | a_m \rangle = \delta_{nm} \\ & \hat{A} = \sum_a a |a\rangle\langle a| && \hat{A} = \int a |a\rangle\langle a| \, da \end{aligned}\]
Measurement / 測定
\[\begin{aligned} & p(a) = \langle \psi | \hat{P}_a | \psi \rangle \\ & \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle \\ & |\psi\rangle \to \frac{\hat{P}_a |\psi\rangle}{\sqrt{\langle \psi | \hat{P}_a | \psi \rangle}} \end{aligned}\]
Evolution / 発展
\[\begin{aligned} & |\psi(t)\rangle = \hat U(t,t_0)\,|\psi(t_0)\rangle \quad \hat U^\dagger(t,t_0)\,\hat U(t,t_0) = \mathbb I \\ & \hat U(t,t_0) = \exp\left[-\frac{i}{\hbar}\,\hat H\,\bigl(t-t_0\bigr)\right] \\ & \hat U(t,t_0) = \mathcal T \exp\left[-\frac{i}{\hbar}\int_{t_0}^{t}\hat H(t')\,dt'\right] \\ \end{aligned}\]
Wave Mechanics / 波動力学
Wave function / 波動関数
\[\begin{aligned} & \langle x | \psi \rangle = \psi(x) \\ & \langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \phi^*(x) \psi(x) dx \\ & \langle x | \hat{x} | \psi \rangle = x \psi(x) \\ & \langle x | \hat{p} | \psi \rangle = -i\hbar \frac{\partial}{\partial x} \psi(x) \end{aligned}\]
Schrödinger equation / Schrödinger方程式
\[\begin{aligned} & -\frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) = i\hbar \frac{\partial\Psi(x,t)}{\partial t} \\ & -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{aligned}\]
Change of basis / 基底変換
\[\begin{aligned} & \langle x | x' \rangle = \delta(x - x') && \langle p | p' \rangle = \delta(p - p') \\ & \langle x | p \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{\frac{i}{\hbar}px} && \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-\frac{i}{\hbar}px} \\ & \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-\frac{i}{\hbar}px} dx && \psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{\frac{i}{\hbar}px} dp \end{aligned}\]
Uncertainty principle / 不確定性原理
\[\begin{aligned} & [\hat{x}, \hat{p}] = i\hbar \\ & \Delta x \Delta p \geq \frac{\hbar}{2} \end{aligned} \\ \psi(x) = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp\left[-\frac{(x-x_0)^2}{4\sigma^2}\right] \exp\left[\frac{i}{\hbar}p_0(x-x_0)\right]\]
Probability current / 確率流密度
\[\begin{aligned} & \frac{\partial |\Psi|^2}{\partial t} = -\nabla \cdot \mathbf{j} \\ & \mathbf{j} = \frac{\hbar}{2mi} (\Psi^* \nabla \Psi - \Psi \nabla \Psi^*) \end{aligned}\]
Ehrenfest’s theorem / Ehrenfestの定理
\[\begin{aligned} & \frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{i\hbar} \langle [\hat{A}, \hat{H}] \rangle \\ & \frac{d}{dt} \langle \hat{x} \rangle = \left\langle \frac{\partial \hat{H}}{\partial \hat{p}} \right\rangle \\ & \frac{d}{dt} \langle \hat{p} \rangle = \left\langle -\frac{\partial \hat{H}}{\partial \hat{x}} \right\rangle \end{aligned}\]
1D System / １次元系
Free Particle / 自由粒子
\[\begin{aligned} & V(x) = 0 \\ & \psi_k(x) = \frac{1}{\sqrt{2\pi}} e^{ikx} \\ & E = \frac{\hbar^2 k^2}{2m} \end{aligned}\]
Step Potential / 段差ポテンシャル
\[\begin{aligned} & V(x) = \begin{cases} 0, & x < 0 \\ V_0, & x \geq 0 \end{cases} \\ & \psi_{k_1}(x) = \begin{cases} \frac{1}{\sqrt{2\pi}} \left( e^{ik_1 x} + \frac{k_1 - k_2}{k_1 + k_2} e^{-ik_1 x} \right), & x < 0 \\ \frac{1}{\sqrt{2\pi}} \frac{2k_1}{k_1 + k_2} e^{ik_2 x}, & x \geq 0 \end{cases} \\ & E = \frac{\hbar^2 k_1^2}{2m} = \frac{\hbar^2 k_2^2}{2m} + V_0 \end{aligned}\]
Square Well / 井戸型ポテンシャル
\[\begin{aligned} & V(x) = \begin{cases} 0, & |x| < \frac{L}{2} \\ \infty, & |x| \geq \frac{L}{2} \end{cases} \\ & \psi_n(x) = \begin{cases} \sqrt{\frac{2}{L}} \cos\left(\frac{n\pi x}{L}\right), & n=1,3,5,\dots \\ \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), & n=2,4,6,\dots \end{cases} \\ & E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} \end{aligned}\]
Harmonic Oscillator / 調和振動子
\[\begin{aligned} & V(x) = \frac{1}{2} m \omega^2 x^2 \\ & \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{m\omega}{\pi\hbar} \right)^{1/4} H_n \left( \sqrt{\frac{m\omega}{\hbar}} x \right) \exp\left( -\frac{m\omega x^2}{2\hbar} \right) \\ & E_n = \hbar \omega \left( n + \frac{1}{2} \right), \quad n = 0, 1, 2, \dots \end{aligned}\]
Ladder Operator / 昇降演算子
\[\begin{aligned} & \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} + \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} \quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} - \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} \\ & \hat{a}|n\rangle = \sqrt{n}|n-1\rangle \quad \hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle \\ & \hat{H}|n\rangle = \hbar\omega\left( n + \frac{1}{2} \right)|n\rangle \end{aligned}\]
Symmetry / 対称性
Spatial Translational Invariance / 空間並進不変性
\[\begin{aligned} & \hat{\mathbf{p}} = -i\hbar\nabla \\ & \hat{U}_\mathbf{a}|\mathbf{r}\rangle = |\mathbf{r}+\mathbf{a}\rangle \to \hat{U}_\mathbf{a} = e^{-\frac{i}{\hbar}\hat{\mathbf{p}}\cdot\mathbf{a}} \\ & \langle\psi_\mathbf{a}|\hat{H}|\psi_\mathbf{a}\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = 0 \end{aligned}\]
Rotational Invariance / 回転不変性
\[\begin{aligned} & \hat{\mathbf{L}} = -i\hbar \mathbf{r} \times \nabla \\ & \hat{U}_\theta|\mathbf{r}\rangle = |R_\theta\mathbf{r}\rangle \to \hat{U}_\theta = e^{-\frac{i}{\hbar}\hat{\mathbf{L}}\cdot\boldsymbol{\theta}} \\ & \langle\psi_{\boldsymbol{\theta}}|\hat{H}|\psi_{\boldsymbol{\theta}}\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{L}}\rangle = 0 \end{aligned}\]
Time Translational Invariance / 時間並進不変性
\[\begin{aligned} & \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}, t) \\ & \hat{U}_\tau|\psi(t)\rangle = |\psi(t+\tau)\rangle \rightarrow \hat{U}_\tau = e^{-\frac{i}{\hbar}\hat{H}\tau} \\ & \langle\psi_{\tau}|\hat{H}|\psi_{\tau}\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{H}\rangle = 0 \end{aligned}\]
Angular Momentum / 角運動量
Angular Momentum Operator / 角運動量演算子
\[\begin{aligned} & [\hat{J}_i, \hat{J}_j] = i\hbar \epsilon_{ijk} \hat{J}_k \quad [\hat{\mathbf{J}}^2, \hat{J}_i] = 0 \\ & \hat{J}_+ = \hat{J}_x + i\hat{J}_y \quad \hat{J}_- = \hat{J}_x - i\hat{J}_y \\ & \hat{J}_{\pm}|j,m\rangle = \hbar\sqrt{(j \mp m)(j \pm m + 1)} |j, m \pm 1\rangle \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=0, \frac{1}{2}, 1, \dots \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \end{aligned}\]
Orbital Angular Momentum / 軌道角運動量
\[\begin{aligned} & \hat{\mathbf{L}}^2 = -\hbar^2 \left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] \\ & \hat{L}_z = -i\hbar\frac{\partial}{\partial\phi} \\ & \hat{\mathbf{L}}^2|l,m\rangle = \hbar^2 l(l+1)|l,m\rangle, \quad l=0, 1, 2, \dots \\ & \hat{L}_z|l,m\rangle = \hbar m|l,m\rangle, \quad m=-l, -l+1, \dots, l \\ & Y_l^m = (-1)^m \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi} \end{aligned}\]
Spin Angular Momentum / スピン角運動量
\[\begin{aligned} & \hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad \hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ & \hat{\mathbf{S}}^2|s,m\rangle = \hbar^2 s(s+1)|s,m\rangle, \quad s=\frac{1}{2} \\ & \hat{S}_z|s,m\rangle = \hbar m|s,m\rangle, \quad m=-\frac{1}{2}, \frac{1}{2} \\ & \hat{H} = -\gamma \hat{\mathbf{S}} \cdot \mathbf{B} \quad \omega_L = \gamma B \end{aligned}\]
Addition of Angular Momentum / 角運動量の合成
\[\begin{aligned} & \hat{\mathbf{J}} = \hat{\mathbf{J}}_1 + \hat{\mathbf{J}}_2 \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=|j_1-j_2|, |j_1-j_2|+1, \dots, j_1+j_2 \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \\ & C_{j_1m_1j_2m_2}^{jm} \neq 0 \to m = m_1 + m_2 \\ & \sum_{m_1,m_2} C_{j_1m_1j_2m_2}^{jm} C_{j_1m_1j_2m_2}^{j'm'} = \delta_{jj'}\delta_{mm'} \end{aligned}\]
3D System / 3次元系
Radial Equation / 動径方程式
\[\begin{aligned} & \psi(r,\theta,\phi) = R(r)Y_l^m(\theta,\phi) \\ & u(r) = rR(r) \\ & -\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[V(r) + \frac{\hbar^2 l(l+1)}{2mr^2}\right]u = Eu \end{aligned}\]
Free Particle / 自由粒子
\[\begin{aligned} & V(r) = 0 \\ & \psi_{klm}(r, \theta, \phi) = \sqrt{\frac{2k^2}{\pi}} j_l(kr) Y_{l}^{m}(\theta, \phi) \\ & E = \frac{\hbar^2 k^2}{2m} \end{aligned}\]
Hydrogen Atom / 水素原子
\[\begin{aligned} & V(r) = -\frac{e^2}{4\pi\epsilon_0 r} \\ & a_0 = \frac{4\pi\epsilon_0\hbar^2}{me^2} \quad \rho = \frac{r}{na_0} \\ & \psi_{nlm}(r,\theta,\phi) = \sqrt{\left(\frac{2}{na_0}\right)^3 \frac{(n-l-1)!}{2n(n+l)!}} e^{-\rho} (2\rho)^l L_{n-l-1}^{2l+1}(2\rho) Y_l^m(\theta,\phi) \\ & E_n = -\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2}\frac{1}{n^2}, \quad n=1,2,3,\dots \\ & L^2 = \hbar^2 l(l+1), \quad l=0,1,\dots,n-1 \\ & L_z = \hbar m, \quad m=-l,-l+1,\dots,l \end{aligned}\]
Electromagnetic Field / 電磁場
\[\begin{aligned} & \hat{H} = \frac{1}{2m}(\hat{\mathbf{p}} - q\mathbf{A})^2 + q\phi \\ & \mathbf{A}' = \mathbf{A} + \nabla\chi, \quad \phi' = \phi - \frac{\partial\chi}{\partial t}, \quad \psi' = e^{\frac{i}{\hbar}q\chi}\psi \\ & \hat{H}\psi = i\hbar\frac{\partial\psi}{\partial t} \to \hat{H}'\psi' = i\hbar\frac{\partial\psi'}{\partial t} \end{aligned}\]
Charged Particle / 荷電粒子
\[\begin{aligned} & E_{n,k_z} = \hbar\omega_c\left(n+\frac{1}{2}\right) + \frac{\hbar^2 k_z^2}{2m}, \quad n=0,1,2,\dots \\ & \omega_c = \frac{|q|B}{m} \\ & \psi = \psi_0 \exp\left(\frac{i}{\hbar}q\int_C \mathbf{A}\cdot d\mathbf{l}\right) \\ & \Delta\varphi = \frac{q}{\hbar}\oint_S \mathbf{B}\cdot d\mathbf{S} \end{aligned}\]