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
\[\begin{aligned}{} & \Delta E = 0 \\ & \Delta S \geq 0 \\ & p_i = \frac{1}{\Omega} \end{aligned}\]
Thermodynamics
Statistical Foundation
Ensemble Theory
Ideal Classical System
Quantum Statistics
Ideal Quantum System
Interacting System
Phase Transition
Non-Equilibrium
Quantum Mechanics / 量子力学
\[\begin{aligned}{} & |\psi\rangle \in \mathcal{H} \\ & \hat{A} = \hat{A}^\dagger \\ & P(a_n) = |\langle a_n | \psi \rangle|^2 \\ & i\hbar \frac{d}{d t} |\psi(t)\rangle = \hat{H}|\psi(t)\rangle \end{aligned}\]
Formalism / 形式論
State / 状態
\[\begin{aligned}{} & \langle n|m\rangle = \delta_{mn}, \quad \langle x|x'\rangle = \delta(x-x') \\ & \sum_{n}|n\rangle\langle n| = I, \quad \int |x\rangle\langle x|\,dx = I \\ & |\psi\rangle = \sum_{n}|n\rangle\langle n|\psi\rangle, \quad |\psi\rangle = \int |x\rangle\langle x|\psi\rangle\,dx \\ & \sum_{n}|\langle n|\psi\rangle|^{2}=1, \quad \int |\langle x|\psi\rangle|^{2}\,dx = 1 \end{aligned}\]
Observable / 観測量
\[\begin{aligned}{} & \hat{A}|a_n\rangle = a_n|a_n\rangle \\ & a_n = a_n^{*} \\ & \langle a_n|a_m\rangle = \delta_{nm} \\ & \hat{A} = \sum_{n}a_n|a_n\rangle\langle a_n| = \int a|a\rangle\langle a|\,da \end{aligned}\]
Measurement / 測定
\[\begin{aligned}{} & \hat{P}_{a}=\sum_{\alpha}|a,\alpha\rangle\langle a,\alpha| \\ & P(a)=\langle\psi|\hat{P}_{a}|\psi\rangle \\ & |\psi\rangle \rightarrow \frac{\hat{P}_{a}|\psi\rangle} {\sqrt{\langle\psi|\hat{P}_{a}|\psi\rangle}} \\ & \langle\hat{A}\rangle=\langle\psi|\hat{A}|\psi\rangle \end{aligned}\]
Evolution / 時間発展
\[\begin{aligned}{} & |\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle \\ & \hat{U}^{\dagger}(t,t_0)\hat{U}(t,t_0)=I \\ & \hat{U}(t,t_0)=\exp\left[-\frac{i}{\hbar}\hat{H}(t-t_0)\right] \\ & \hat{U}(t,t_0)=\mathcal{T}\exp\left[-\frac{i}{\hbar} \int_{t_0}^{t}\hat{H}(t')\,dt'\right] \end{aligned}\]
Wave Mechanics / 波動力学
Wave Function / 波動関数
\[\begin{aligned}{} & \psi(\mathbf{r},t) = \langle \mathbf{r}|\psi(t)\rangle \\ & \phi(\mathbf{p},t) = \langle \mathbf{p}|\psi(t)\rangle \\ & \psi(\mathbf{r}) = \frac{1}{(2\pi\hbar)^{3/2}}\int \phi(\mathbf{p})e^{\frac{i}{\hbar}\mathbf{p}\cdot\mathbf{r}}\,d^{3}p \\ & \phi(\mathbf{p}) = \frac{1}{(2\pi\hbar)^{3/2}}\int \psi(\mathbf{r})e^{-\frac{i}{\hbar}\mathbf{p}\cdot\mathbf{r}}\,d^{3}r \end{aligned}\]
Operator / 演算子
\[\begin{aligned}{} & \hat{\mathbf{r}}\psi(\mathbf{r}) = \mathbf{r}\psi(\mathbf{r}), \quad \hat{\mathbf{p}}\psi(\mathbf{r}) = -i\hbar\nabla\psi(\mathbf{r}) \\ & \hat{\mathbf{p}}\phi(\mathbf{p}) = \mathbf{p}\phi(\mathbf{p}), \quad \hat{\mathbf{r}}\phi(\mathbf{p}) = i\hbar\nabla_{\mathbf{p}}\phi(\mathbf{p}) \\ & [\hat{x}_i,\hat{p}_j] = i\hbar\delta_{ij}, \quad [\hat{x}_i,\hat{x}_j] = 0, \quad [\hat{p}_i,\hat{p}_j] = 0 \end{aligned}\]
Expectation Value / 期待値
\[\begin{aligned}{} & \langle \hat{A}\rangle = \int \psi^{*}(\mathbf{r})\hat{A}\psi(\mathbf{r})\,d^{3}r \\ & \Delta A\Delta B \geq \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle\right| \\ & \Delta x\Delta p \geq \frac{\hbar}{2} \end{aligned}\]
Schrödinger Equation / Schrödinger方程式
\[\begin{aligned}{} & \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},t)\right]\psi(\mathbf{r},t) = i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \\ & \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})\right]u(\mathbf{r}) = Eu(\mathbf{r}) \\ & \psi(\mathbf{r},t)=\sum_{n}c_{n}u_{n}(\mathbf{r})e^{-\frac{i}{\hbar}E_{n}t} \end{aligned}\]
Probability Current / 確率流
\[\begin{aligned}{} & \rho(\mathbf{r},t)=|\psi(\mathbf{r},t)|^{2} \\ & \mathbf{j}(\mathbf{r},t)=\frac{\hbar}{2mi}\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{*}\right) \\ & \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=0 \end{aligned}\]
Ehrenfest Theorem / Ehrenfestの定理
\[\begin{aligned}{} & \frac{d}{dt}\langle\hat{A}\rangle = \frac{1}{i\hbar}\langle[\hat{A},\hat{H}]\rangle +\left\langle\frac{\partial\hat{A}}{\partial t}\right\rangle \\ & \frac{d}{dt}\langle\hat{\mathbf{r}}\rangle = \frac{\langle\hat{\mathbf{p}}\rangle}{m} \\ & \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = -\langle\nabla V\rangle \end{aligned}\]
1D System / 一次元系
Boundary Conditions / 境界条件
\[\begin{aligned}{} & -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \\ & V(x_0) < \infty \rightarrow \psi(x_0^+) = \psi(x_0^-), \quad \psi'(x_0^+) = \psi'(x_0^-) \\ & V(x_0) = \infty \rightarrow \psi(x_0^+) = \psi(x_0^-) \end{aligned}\]
Free Particle / 自由粒子
\[\begin{aligned}{} & V(x) = 0 \\ & \psi(x) = A e^{ikx} + B e^{-ikx} \\ & k = \frac{\sqrt{2mE}}{\hbar} \end{aligned}\]
Potential Step / ポテンシャル段差
\[\begin{aligned}{} & V(x) = \begin{cases} 0, & x < 0 \\ V_0, & x \geq 0 \end{cases} \\ & \psi(x) = \begin{cases} A e^{ik_1 x} + B e^{-ik_1 x}, & x < 0 \\ C e^{ik_2 x}, & x \geq 0 \end{cases} \\ & E > V_0: \quad k_1 = \frac{\sqrt{2mE}}{\hbar}, \quad k_2 = \frac{\sqrt{2m(E-V_0)}}{\hbar} \\ & R = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2, \quad T = \frac{4k_1 k_2}{(k_1 + k_2)^2} \end{aligned}\]
Potential Barrier / ポテンシャル障壁
\[\begin{aligned}{} & V(x) = \begin{cases} 0, & x < 0 \\ V_0, & 0 \leq x \leq a \\ 0, & x > a \end{cases} \\ & \psi(x) = \begin{cases} A e^{ikx} + B e^{-ikx}, & x < 0 \\ C e^{\kappa x} + D e^{-\kappa x}, & 0 \leq x \leq a \\ F e^{ikx}, & x > a \end{cases} \\ & E < V_0: \quad k = \frac{\sqrt{2mE}}{\hbar}, \quad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar} \\ & R = \left[ 1 + \frac{4E(V_0-E)}{V_0^2 \sinh^2(\kappa a)} \right]^{-1}, \quad T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1} \end{aligned}\]
Square Well / 井戸型ポテンシャル
\[\begin{aligned}{} & V(x) = \begin{cases} 0, & 0 < x < L \\ \infty, & x\leq 0,\; x\geq L \end{cases} \\ & \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)\\ & E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2}, \quad n=1,2,3,\dots \end{aligned}\]
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 Symmetry / 空間並進対称性
\[\begin{aligned}{} & \hat{\mathbf{p}} = -i\hbar\nabla \\ & \hat{U}_{\mathbf{a}} = e^{-\frac{i}{\hbar}\mathbf{a}\cdot\hat{\mathbf{p}}} \to \hat{U}_{\mathbf{a}}\psi(\mathbf{r}) = \psi(\mathbf{r}-\mathbf{a}) \\ & \langle \hat{U}_{\mathbf{a}}\psi|\hat{H}|\hat{U}_{\mathbf{a}}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = 0 \end{aligned}\]
Rotational Symmetry / 回転対称性
\[\begin{aligned}{} & \hat{\mathbf{L}} = -i\hbar\mathbf{r}\times\nabla \\ & \hat{U}_{\boldsymbol{\theta}} = e^{-\frac{i}{\hbar}\boldsymbol{\theta}\cdot\hat{\mathbf{L}}} \to \hat{U}_{\boldsymbol{\theta}}\psi(\mathbf{r}) = \psi(R_{\boldsymbol{\theta}}^{-1}\mathbf{r}) \\ & \langle \hat{U}_{\boldsymbol{\theta}}\psi|\hat{H}|\hat{U}_{\boldsymbol{\theta}}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{L}}\rangle = 0 \end{aligned}\]
Time Translational Symmetry / 時間並進対称性
\[\begin{aligned}{} & \hat{E} = i\hbar\frac{\partial}{\partial t} \\ & \hat{U}_{\tau} = e^{-\frac{i}{\hbar}\tau\hat{E}} \to \hat{U}_{\tau}\psi(t) = \psi(t+\tau) \\ & \langle \hat{U}_{\tau}\psi|\hat{H}|\hat{U}_{\tau}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{E}\rangle = 0 \end{aligned}\]
Parity Symmetry / パリティ対称性
\[\begin{aligned}{} & \hat{\Pi}\psi(\mathbf{r}) = \psi(-\mathbf{r}), \quad \pi = \pm 1 \\ & \langle \hat{\Pi}\psi|\hat{H}|\hat{\Pi}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\Pi}\rangle = 0 \end{aligned}\]
Angular Momentum / 角運動量
Angular Momentum Operator / 角運動量演算子
\[\begin{aligned}{} & [\hat{J}_i, \hat{J}_j] = i\hbar \epsilon_{ijk} \hat{J}_k, \quad [\hat{\mathbf{J}}^2, \hat{J}_i] = 0 \\ & \hat{J}_+ = \hat{J}_x + i\hat{J}_y, \quad \hat{J}_- = \hat{J}_x - i\hat{J}_y \\ & \hat{J}_{\pm}|j,m\rangle = \hbar\sqrt{j(j+1) - m(m \pm 1)} |j, m \pm 1\rangle \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=0, \frac{1}{2}, 1, \dots \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \end{aligned}\]
Orbital Angular Momentum / 軌道角運動量
\[\begin{aligned}{} & \hat{\mathbf{L}} = -i\hbar \mathbf{r} \times \nabla \to [\hat{L}_i, \hat{L}_j] = i\hbar \epsilon_{ijk} \hat{L}_k\\ & \hat{\mathbf{L}}^2 = -\hbar^2 \left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] \\ & \hat{L}_z = -i\hbar\frac{\partial}{\partial\phi} \\ & \hat{\mathbf{L}}^2|l,m\rangle = \hbar^2 l(l+1)|l,m\rangle, \quad l=0, 1, 2, \dots \\ & \hat{L}_z|l,m\rangle = \hbar m|l,m\rangle, \quad m=-l, -l+1, \dots, l \\ & Y_l^m(\theta, \phi) = (-1)^m \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi} \end{aligned}\]
Spin Angular Momentum / スピン角運動量
\[\begin{aligned}{} & \hat{\mathbf{S}} = \frac{\hbar}{2} \boldsymbol{\sigma} \to [\hat{S}_i, \hat{S}_j] = i\hbar \epsilon_{ijk} \hat{S}_k\\ & \hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ & \hat{\mathbf{S}}^2|s,m\rangle = \hbar^2 s(s+1)|s,m\rangle, \quad s=\frac{1}{2} \\ & \hat{S}_z|s,m\rangle = \hbar m|s,m\rangle, \quad m=-\frac{1}{2}, \frac{1}{2} \\ & \hat{H} = -\gamma \hat{\mathbf{S}} \cdot \mathbf{B}, \quad \omega_L = \gamma B \end{aligned}\]
Addition of Angular Momentum / 角運動量の合成
\[\begin{aligned}{} & \hat{\mathbf{J}} = \hat{\mathbf{J}}_1 + \hat{\mathbf{J}}_2 \to [\hat{J}_i, \hat{J}_j] = i\hbar \epsilon_{ijk} \hat{J}_k \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=|j_1-j_2|, |j_1-j_2|+1, \dots, j_1+j_2 \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \\ & |j_1,j_2;j,m\rangle = \sum_{m_1,m_2} C_{j_1m_1j_2m_2}^{jm} |j_1,m_1\rangle |j_2,m_2\rangle \\ & C_{j_1m_1j_2m_2}^{jm} \neq 0 \to m = m_1 + m_2 \\ & \sum_{m_1,m_2} C_{j_1m_1j_2m_2}^{jm} C_{j_1m_1j_2m_2}^{j'm'} = \delta_{jj'}\delta_{mm'} \end{aligned}\]
3D System / 三次元系
Central Potential / 中心力ポテンシャル
\[\begin{aligned}{} & -\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r}) \\ & V(\mathbf{r}) = V(r), \quad \psi(\mathbf{r}) = R(r)Y_l^m(\theta,\phi), \quad u(r) = rR(r) \\ & -\frac{\hbar^2}{2m}\frac{d^2u(r)}{dr^2} + \left[V(r) + \frac{\hbar^2 l(l+1)}{2mr^2}\right]u(r) = Eu(r) \end{aligned}\]
Infinite Spherical Well / 無限球形井戸
\[\begin{aligned}{} & V(r) = \begin{cases} 0, & r < a \\ \infty, & r \geq a \end{cases} \\ & \psi_{nlm}(r,\theta,\phi) = \frac{\sqrt{2}}{a^{3/2}\left|j_{l+1}(\alpha_{nl})\right|} j_l(k_{nl}r)Y_l^m(\theta,\phi) \\ & j_l(\alpha_{nl})=0, \quad k_{nl} = \frac{\alpha_{nl}}{a} \\ & E_{nl} = \frac{\hbar^2 k_{nl}^2}{2m}, \quad n = 1,2,3,\dots, \quad l = 0,1,2,\dots \end{aligned}\]
Isotropic Harmonic Oscillator / 等方性調和振動子
\[\begin{aligned}{} & V(r) = \frac{1}{2}m\omega^2 r^2 \\ & \psi_{n_x n_y n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ & N = n_x+n_y+n_z, \quad g_N = \frac{(N+1)(N+2)}{2} \\ & E_N = \hbar\omega\left(N+\frac{3}{2}\right), \quad N = 0,1,2,\dots \end{aligned}\]
Hydrogen Atom / 水素原子
\[\begin{aligned}{} & V(r) = -\frac{e^2}{4\pi\epsilon_0 r} \\ & \psi_{nlm}(r,\theta,\phi) = \sqrt{\left(\frac{2}{na_0}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]}} e^{-\rho} (2\rho)^l L_{n-l-1}^{2l+1}(2\rho) Y_l^m(\theta,\phi) \\ & a_0 = \frac{4\pi\epsilon_0\hbar^2}{me^2}, \quad \rho = \frac{r}{na_0} \\ & E_n = -\frac{me^4}{2(4\pi\varepsilon_0)^2\hbar^2}\frac{1}{n^2}, \quad n=1,2,3,\dots \end{aligned}\]
Electromagnetic Potential / 電磁ポテンシャル
\[\begin{aligned}{} & \hat{H}\psi = \left[\frac{1}{2m}(\hat{\mathbf{p}} - q\mathbf{A})^2 + q\Phi\right]\psi = i\hbar\frac{\partial\psi}{\partial t} \\ & \mathbf{A}' = \mathbf{A} + \nabla\chi, \quad \Phi' = \Phi - \frac{\partial\chi}{\partial t}, \quad \psi' = e^{\frac{i}{\hbar}q\chi}\psi \\ & \hat{H'}\psi' = \left[\frac{1}{2m}(\hat{\mathbf{p}} - q\mathbf{A'})^2 + q\Phi'\right]\psi' = i\hbar\frac{\partial\psi'}{\partial t} \\ \end{aligned}\]
Charged Particle / 荷電粒子
\[\begin{aligned}{} & E_{n,k_z} = \hbar\omega_c\left(n+\frac{1}{2}\right) + \frac{\hbar^2 k_z^2}{2m}, \quad \omega_c=\frac{|q|B}{m} \\ & \psi = \psi_0 \exp\left(\frac{i}{\hbar}q\int_C \mathbf{A}\cdot d\mathbf{l}\right),\quad \Delta\varphi = \frac{q}{\hbar}\int_S \mathbf{B}\cdot d\mathbf{S} \end{aligned}\]
Approximation Method / 近似法
Variational Method / 変分法
\[\begin{aligned}{} & E_0 \leq \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle} \end{aligned}\]
WKB Approximation / WKB近似
\[\begin{aligned}{} & \psi(x) \approx \frac{1}{\sqrt{p(x)}} \left( C_1 e^{\frac{i}{\hbar}\int^x p(x')\,dx'} + C_2 e^{-\frac{i}{\hbar}\int^x p(x')\,dx'} \right) \\ & \psi(x) \approx \frac{1}{\sqrt{\kappa(x)}} \left( C_1 e^{\frac{1}{\hbar}\int^x \kappa(x')\,dx'} + C_2 e^{-\frac{1}{\hbar}\int^x \kappa(x')\,dx'} \right) \end{aligned}\]
Bound State and Tunneling / 束縛状態とトンネル効果
\[\begin{aligned}{} & \int_{x_1}^{x_2} p(x)\,dx = \left(n+\frac{1}{2}\right)\pi\hbar \\ & T = \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \kappa(x)\,dx\right) \end{aligned}\]
Perturbation Theory / 摂動論
Non-Degenerate Perturbation Theory / 非縮退定常摂動論
\[\begin{aligned}{} & E_n^{(1)} = \langle \psi_n^{(0)} | \hat{V} | \psi_n^{(0)} \rangle \\ & |\psi_n^{(1)}\rangle = \sum_{m \neq n} \frac{|\psi_m^{(0)}\rangle \langle \psi_m^{(0)} | \hat{V} | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} \\ & E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | \hat{V} | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}} \end{aligned}\]
Degenerate Perturbation Theory / 縮退定常摂動論
\[\begin{aligned}{} & E_{n,\alpha}^{(1)} : \sum_j \langle \psi_{n,i}^{(0)} | \hat{V} | \psi_{n,j}^{(0)} \rangle C_{n,\alpha}^j = E_{n,\alpha}^{(1)} C_{n,\alpha}^i \\ & |\psi_{n,\alpha}^{(1)}\rangle = \sum_{m \neq n, \beta} \frac{|\psi_{m,\beta}^{(0)}\rangle \langle \psi_{m,\beta}^{(0)} | \hat{V} | \psi_{n,\alpha}^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} \\ & E_{n,\alpha}^{(2)} = \sum_{m \neq n, \beta} \frac{|\langle \psi_{m,\beta}^{(0)} | \hat{V} | \psi_{n,\alpha}^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}} \end{aligned}\]
Time-Dependent Perturbation Theory / 時間依存摂動論
\[\begin{aligned}{} & i\hbar \frac{\partial C_n(t)}{\partial t} = \sum_m \langle \psi_n^{(0)} | \hat{V}(t) | \psi_m^{(0)} \rangle e^{\frac{i}{\hbar}(E_n - E_m)t} C_m(t) \\ & C_n^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle \psi_n^{(0)} | \hat{V}(t') | \psi_i^{(0)} \rangle e^{\frac{i}{\hbar}(E_n - E_i)t'} \, dt' \\ & \hat{V}(t) = \hat{V} e^{\eta t}, \; \eta \to 0 : \quad C_n^{(1)}(0) = \frac{\langle \psi_n^{(0)} | \hat{V} | \psi_i^{(0)} \rangle}{E_i - E_n} \\ & \hat{V}(t) = \hat{V} e^{-i\omega t}, \; t \ge 0 : \quad C_n^{(1)}(t) = \frac{\langle \psi_n^{(0)} | \hat{V} | \psi_i^{(0)} \rangle}{E_n - E_i - \hbar\omega} \left[ 1 - e^{\frac{i}{\hbar}(E_n - E_i - \hbar\omega)t} \right] \end{aligned}\]
Scattering Theory / 散乱理論
Lippmann-Schwinger Equation / Lippmann-Schwinger方程式
\[\begin{aligned}{} & |\psi^{(+)}\rangle = |\psi^{(0)}\rangle + \frac{1}{E-\hat{H}_0+i\epsilon}\hat{V}|\psi^{(+)}\rangle \\ & \psi^{(+)}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} -\frac{m}{2\pi\hbar^2} \int V(\mathbf{r}') \psi^{(+)}(\mathbf{r}') \frac{e^{ik|\mathbf{r}-\mathbf{r}'|}} {|\mathbf{r}-\mathbf{r}'|} \,d^3r' \end{aligned}\]
Scattering Amplitude / 散乱振幅
\[\begin{aligned}{} & r\to\infty,\quad V(\mathbf{r})\to 0 \\ & f(\mathbf{k}_f \leftarrow \mathbf{k}_i) = -\frac{m}{2\pi\hbar^2} \int V(\mathbf{r}) \psi_{\mathbf{k}_i}^{(+)}(\mathbf{r}) e^{-i\mathbf{k}_f\cdot\mathbf{r}} \,d^3r \\ & \psi^{(+)}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} + f(\theta,\phi)\frac{e^{ikr}}{r} \\ & \frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2 \end{aligned}\]
Born Approximation / Born近似
\[\begin{aligned}{} & \psi_{\mathbf{k}_i}^{(+)}(\mathbf{r}) \approx e^{i\mathbf{k}_i\cdot\mathbf{r}}, \quad \mathbf{q}=\mathbf{k}_f-\mathbf{k}_i \\ & f(\mathbf{q}) = -\frac{m}{2\pi\hbar^2} \int V(\mathbf{r}) e^{-i\mathbf{q}\cdot\mathbf{r}} \,d^3r \end{aligned}\]
Partial Wave Expansion / 部分波展開
\[\begin{aligned}{} & V(\mathbf{r}) = V(r) \\ & f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^{\infty} (2\ell+1) \left(e^{2i\delta_\ell}-1\right) P_\ell(\cos\theta) \\ & \sigma_{\mathrm{tot}} = \frac{4\pi}{k}\operatorname{Im} f(0) \end{aligned}\]
Path Integral / 経路積分
Configuration Space / 座標空間
\[\begin{aligned}{} & U(x_f, t_f; x_i, t_i) = \int \mathcal{D}[x] e^{\frac{i}{\hbar}S[x]} \\ & \int \mathcal{D}[x] = \lim_{N \to \infty} \left( \frac{m}{2\pi i \hbar \Delta t} \right)^{\frac{N}{2}} \prod_{j=1}^{N-1} dx_j \\ & S[x] = \sum_{j=1}^{N} \left[ \frac{m}{2} \left( \frac{x_j - x_{j-1}}{\Delta t} \right)^2 - V(x_j) \right] \Delta t \end{aligned}\]
Phase Space / 位相空間
\[\begin{aligned}{} & U(x_f, t_f; x_i, t_i) = \int \mathcal{D}[x]\mathcal{D}[p] e^{\frac{i}{\hbar}S[x,p]} \\ & \int \mathcal{D}[x]\mathcal{D}[p] = \lim_{N \to \infty} \left( \frac{1}{2\pi \hbar} \right)^N \prod_{j=1}^{N-1} dx_j \prod_{k=1}^N dp_k \\ & S[x,p] = \sum_{j=1}^{N} \left[ p_j \left( \frac{x_j - x_{j-1}}{\Delta t} \right) - H(x_j, p_j) \right] \Delta t \end{aligned}\]
Free Particle / 自由粒子
\[\begin{aligned}{} & S_{cl} = \frac{m(x_f - x_i)^2}{2T} \\ & U(x_f, T; x_i, 0) = \sqrt{\frac{m}{2\pi i \hbar T}} \exp\left( \frac{i m (x_f - x_i)^2}{2 \hbar T} \right) \end{aligned}\]
Harmonic Oscillator / 調和振動子
\[\begin{aligned}{} & S_{cl} = \frac{m\omega}{2\sin(\omega T)} \left[ (x_i^2 + x_f^2)\cos(\omega T) - 2x_i x_f \right] \\ & U(x_f, T; x_i, 0) = \sqrt{\frac{m\omega}{2\pi i \hbar \sin(\omega T)}} \exp\left( \frac{i m \omega}{2\hbar \sin(\omega T)} \left[ (x_i^2 + x_f^2)\cos(\omega T) - 2x_i x_f \right] \right) \end{aligned}\]
Identical Particle / 同種粒子
Exchange Operator / 交換演算子
\[\begin{aligned}{} & \hat{P}_{ij} (\dots |w_i\rangle \dots |w_j\rangle \dots) = (\dots |w_j\rangle \dots |w_i\rangle \dots) \\ & \begin{aligned}{} & \text{Boson:} && \hat{P}_{ij} |\psi\rangle = |\psi\rangle \\ & \text{Fermion:} && \hat{P}_{ij} |\psi\rangle = -|\psi\rangle \\ & && w_i = w_j \rightarrow |\psi\rangle = 0 \end{aligned} \end{aligned}\]
First Quantization / 第一量子化
\[\begin{aligned}{} & \text{Boson:} && |w_1 \le w_2 \le \dots \le w_N\rangle = \frac{1}{\sqrt{N! \prod_\alpha n_\alpha!}} \sum_{\sigma \in S_N} |w_{\sigma(1)}\rangle |w_{\sigma(2)}\rangle \dots |w_{\sigma(N)}\rangle \\ & \text{Fermion:} && |w_1 < w_2 < \dots < w_N\rangle = \frac{1}{\sqrt{N!}} \sum_{\sigma \in S_N} \text{sgn}(\sigma) |w_{\sigma(1)}\rangle |w_{\sigma(2)}\rangle \dots |w_{\sigma(N)}\rangle \end{aligned}\]
Creation & Annihilation Operator / 生成・消滅演算子
\[\begin{aligned}{} & \text{Boson:} && a_\alpha^\dagger |\dots, n_\alpha, \dots\rangle = \sqrt{n_\alpha + 1} |\dots, n_\alpha + 1, \dots\rangle \\ & && a_\alpha |\dots, n_\alpha, \dots\rangle = \sqrt{n_\alpha} |\dots, n_\alpha - 1, \dots\rangle \\ & && [a_\alpha, a_\beta^\dagger] = \delta_{\alpha\beta} \quad [a_\alpha^\dagger, a_\beta^\dagger] = 0 \\ & \text{Fermion:} && a_\alpha^\dagger |\dots, n_\alpha, \dots\rangle = (-1)^{\sum_{\beta < \alpha} n_\beta} (1 - n_\alpha) |\dots, 1, \dots\rangle \\ & && a_\alpha |\dots, n_\alpha, \dots\rangle = (-1)^{\sum_{\beta < \alpha} n_\beta} n_\alpha |\dots, 0, \dots\rangle \\ & && \{a_\alpha, a_\beta^\dagger\} = \delta_{\alpha\beta} \quad \{a_\alpha^\dagger, a_\beta^\dagger\} = 0 \\ & && (a_\alpha^\dagger)^2 = 0 \end{aligned}\]
Second Quantization / 第二量子化
\[\begin{aligned}{} & \text{Boson:} && |n_1, n_2, \dots, n_N\rangle = \prod_\alpha \frac{1}{\sqrt{n_\alpha!}} (a_1^\dagger)^{n_1} (a_2^\dagger)^{n_2} \dots (a_N^\dagger)^{n_N} |0\rangle \\ & \text{Fermion:} && |n_1, n_2, \dots, n_N\rangle = (a_1^\dagger)^{n_1} (a_2^\dagger)^{n_2} \dots (a_N^\dagger)^{n_N} |0\rangle \end{aligned}\]
Dirac Equation / Dirac方程式
Relativistic Wave Equation / 相対論的波動方程式
\[\begin{aligned}{} & \hat{H}^2 = \hat{p}^{\,2}c^2 + m^2c^4 \\ & \left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 + \frac{m^2c^2}{\hbar^2}\right)\psi = 0 \\ & \hat{H} = c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2 \\ & \left(i\hbar\gamma^\mu\partial_\mu - mc\right)\psi = 0 \end{aligned}\]
Dirac Representation / Dirac表示
\[\begin{aligned}{} & \{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}, \quad \gamma^0 = \begin{pmatrix} \mathbb{I} & 0 \\ 0 & -\mathbb{I} \end{pmatrix} ,\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} \\ & \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} ,\quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} ,\quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{aligned}\]
Plane Wave Solution / 平面波解
\[\begin{aligned}{} & \psi(x) = u(p)e^{-\frac{i}{\hbar}p_\mu x^\mu}, && \psi(x) = v(p)e^{\frac{i}{\hbar}p_\mu x^\mu} \\ & u(p) = \begin{pmatrix} \chi^s \\ \frac{c\boldsymbol{\sigma}\cdot\mathbf{p}}{E+mc^2}\chi^s \end{pmatrix}, && v(p) = \begin{pmatrix} \frac{c\boldsymbol{\sigma}\cdot\mathbf{p}}{E+mc^2}\eta^s \\ \eta^s \end{pmatrix} \\ & E = \sqrt{p^2c^2 + m^2c^4}, && -E = -\sqrt{p^2c^2 + m^2c^4} \end{aligned}\]
Non-Relativistic Limit / 非相対論的極限
\[\begin{aligned}{} & \psi = e^{-\frac{i}{\hbar}mc^2t} \begin{pmatrix} \varphi \\ \chi \end{pmatrix} \\ & \chi \approx \frac{\boldsymbol{\sigma}\cdot(\hat{\mathbf{p}}-q\mathbf{A})}{2mc}\varphi \\ & i\hbar\frac{\partial\varphi}{\partial t} = \left[ \frac{(\hat{\mathbf{p}}-q\mathbf{A})^2}{2m} + q\Phi -\frac{q\hbar}{2m}\boldsymbol{\sigma}\cdot\mathbf{B} \right]\varphi \end{aligned}\]
Conserved Current / 保存カレント
\[\begin{aligned}{} & j^\mu = c\psi^\dagger\gamma^0\gamma^\mu\psi \\ & \partial_\mu j^\mu = 0 \end{aligned}\]