Physics 🍊⭐

Published: January 02, 2023

Equations of Physics

Classical Mechanics / 古典力学

Newtonian Mechanics / Newton力学

Equation of Motion / 運動方程式

\[\mathbf{F} = \frac{d}{dt}\left(m\mathbf{v}\right) \\ \mathbf{F}_{\alpha\beta} = -\mathbf{F}_{\beta\alpha}\]

Momentum / 運動量

\[\frac{d\mathbf{P}}{dt} = \mathbf{F} \\ \begin{aligned}{} & \mathbf{P} = m\mathbf{v} && \mathbf{F} \\ & \mathbf{P} = M\mathbf{V} && \mathbf{F} = \sum_{\alpha}\mathbf{F}_\alpha^{(e)} \end{aligned}\]

\[\frac{d\mathbf{P}}{dt} = \mathbf{F} \\ \begin{aligned}{} \mathbf{P} &= \sum_\alpha m_\alpha\left(\mathbf{V} + \mathbf{v'}_\alpha\right) \\ &= \sum_\alpha m_\alpha\mathbf{V} + \sum_\alpha m_\alpha\mathbf{v'}_\alpha \\ &= M\mathbf{V} \end{aligned} \\ \begin{aligned}{} \mathbf{F} &= \sum_\alpha\left(\mathbf{F}_\alpha^{(e)} + \sum_{\beta\neq\alpha}\mathbf{F}_{\alpha\beta}\right) \\ &= \sum_\alpha\mathbf{F}_{\alpha}^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{F}_{\alpha\beta} + \mathbf{F}_{\beta\alpha}\right) \\ &= \sum_\alpha\mathbf{F}_\alpha^{(e)} \end{aligned}\]

Angular Momentum / 角運動量

\[\frac{d\mathbf{M}}{dt} = \mathbf{N} \\ \begin{aligned}{} & \mathbf{M} = m\mathbf{r}\times\mathbf{v} && \mathbf{N} = \mathbf{r}\times\mathbf{F} \\ & \mathbf{M} = M\mathbf{R}\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha && \mathbf{N} = \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} \end{aligned}\]

\[\frac{d\mathbf{M}}{dt} = m\mathbf{v}\times\mathbf{v} + m\mathbf{r}\times\mathbf{a} = \mathbf{r}\times\mathbf{F} = \mathbf{N} \\ \begin{aligned}{} \mathbf{M} &= \sum_\alpha m_\alpha\left(\mathbf{R}+\mathbf{r'}_\alpha\right)\times\left(\mathbf{V}+\mathbf{v'}_\alpha\right) \\ &= \sum_\alpha m_\alpha\mathbf{R}\times\mathbf{V} + \mathbf{R}\times\left(\sum_\alpha m_\alpha\mathbf{v}'_\alpha\right) \\ &+ \left(\sum_\alpha m_\alpha\mathbf{r'}_\alpha\right)\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha \\ &= M\mathbf{R}\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha \end{aligned} \\ \begin{aligned}{} \mathbf{N} &= \sum_\alpha \mathbf{r}_\alpha\times\left(\mathbf{F}_\alpha^{(e)} + \sum_{\beta\neq\alpha}\mathbf{F_{\alpha\beta}}\right) \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{r}_\alpha\times\mathbf{F}_{\alpha\beta} + \mathbf{r}_\beta\times\mathbf{F}_{\beta\alpha}\right) \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{r}_\alpha-\mathbf{r}_\beta\right)\times\mathbf{F}_{\alpha\beta} \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} \end{aligned}\]

Energy / エネルギー

\[\frac{dE}{dt} = 0 \\ \begin{aligned}{} E &= \frac{1}{2} mv^2 + U \\ E &= \frac{1}{2} MV^2 + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 + U \\ \end{aligned}\]

\[\frac{dE}{dt} = m\mathbf{v} \cdot \mathbf{a} + (\nabla U) \cdot \mathbf{v} + \frac{\partial U}{\partial t} = (\mathbf{F} + \nabla U) \cdot \mathbf{v} = 0 \\ \begin{aligned}{} E &= \sum_\alpha \frac{1}{2} m_\alpha (\mathbf{V} + \mathbf{v}_\alpha') \cdot (\mathbf{V} + \mathbf{v}_\alpha') \\ &= \sum_\alpha \frac{1}{2} m_\alpha V^2 + \left(\sum_\alpha m_\alpha \mathbf{v}_\alpha'\right) \cdot \mathbf{V} + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 \\ &= \frac{1}{2} MV^2 + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 \end{aligned}\]

Lagrangian Mechanics / Lagrange力学

Lagrange’s Equation / Langrange方程式

\[\delta S = \delta\int_{t_1}^{t_2} L dt = 0 \quad L = T - U\\ \begin{aligned}{} & \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} = 0 \\ & \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} + \sum_j \lambda_j \frac{\partial{f_j}}{\partial{q_i}} = 0 \end{aligned}\]

\[\begin{aligned}{} \delta S &= \int_{t_1}^{t_2} \sum_i \left( \frac{\partial{L}}{\partial{q_i}} \delta q_i + \frac{\partial{L}}{\partial{\dot{q}_i}} \delta \dot{q}_i \right) dt \\ &= \int_{t_1}^{t_2} \sum_i \frac{\partial{L}}{\partial{q_i}} \delta q_i dt + \left[ \sum_i \frac{\partial{L}}{\partial{\dot{q}_i}} \delta q_i \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \sum_i \frac{d}{dt} \left( \frac{\partial{L}}{\partial{\dot{q_i}}} \right) \delta q_i dt \\ &= \int_{t_1}^{t_2} \sum_i \left( \frac{\partial{L}}{\partial{q_i}} - \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} \right) \delta q_i dt \\ &= 0 \end{aligned}\]

Conservation Law / 保存則

\[\begin{aligned}{} E &= \sum_i \dot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} - L = const. \\ \mathbf{P} &= \sum_\alpha \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} = const. \\ \mathbf{M} &= \sum_\alpha \mathbf{r}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} = const. \end{aligned}\]

\[\begin{aligned}{} & \quad\; \frac{d}{dt} \left( \sum_i \dot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} - L \right) \\ &= \sum_i \left[ \ddot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} + \dot{q}_i \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} \dot{q}_i - \frac{\partial{L}}{\partial{\dot{q}_i}} \ddot{q}_i \right] \\ &= 0 \\ \delta L &= \sum_\alpha \frac{\partial{L}}{\partial{\mathbf{r}_\alpha}} \cdot \mathbf{\epsilon} \\ &= \mathbf{\epsilon} \cdot \sum_\alpha \frac{d}{dt} \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \\ &= 0 \\ \delta L &= \sum_\alpha \left( \frac{\partial{L}}{\partial{\mathbf{r}_\alpha}} \cdot \mathbf{\theta} \times \mathbf{r}_\alpha + \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \cdot \mathbf{\theta} \times \mathbf{v}_\alpha \right) \\ &= \sum_\alpha \left[ \mathbf{\theta} \cdot \mathbf{r}_\alpha \times \frac{d}{dt}\frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} + \mathbf{\theta} \cdot \mathbf{v}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \right] \\ &= \mathbf{\theta} \cdot \sum_\alpha \frac{d}{dt} \left( \mathbf{r}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \right) \\ &= 0 \end{aligned}\]

Galilean Transformation / ガリレイ変換

\[\begin{aligned}{} \mathbf{r}' &= \mathbf{r} + \mathbf{V}t \\ t' &= t \end{aligned}\]

\[\begin{aligned}{} L' &= \frac{1}{2} m \left( \mathbf{v}' \right)^2 - U'(\mathbf{r}') \\ &= \frac{1}{2} m \mathbf{v}^2 + m \mathbf{v} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 - U(\mathbf{r}) \\ &= L + \frac{d}{dt} \left( m \mathbf{r} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 t \right) \\ \delta S' &= \delta \int_{t_1}^{t_2} L' dt' \\ &= \delta \int_{t_1}^{t_2} L dt + \delta \left[ m \mathbf{r} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 t \right]_{t_1}^{t_2} \\ &= \delta S \end{aligned}\]

Virial Theorem / Virial定理

\[U(\alpha \mathbf{r}) = \alpha ^k U(\mathbf{r}) \\ \frac{t'}{t} = \left( \frac{l'}{l} \right)^{1-\frac{k}{2}} \\ \langle T \rangle = \frac{k}{2}\langle U \rangle\]

\[l' = \alpha l \quad t' = \beta t \\ \frac{T'}{T} = \alpha^2 \beta^{-2} = \frac{U'}{U} = \alpha^k \\ \beta = \alpha^{1-\frac{k}{2}} \\ \begin{aligned}{} 2T &= \sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{v}_\alpha \\ &= \frac{d}{dt}\left(\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right) - \sum_\alpha\mathbf{F}_\alpha\cdot\mathbf{r}_\alpha \end{aligned} \\ \begin{aligned}{} \langle 2T \rangle &= \lim_{\tau\to\infty}\frac{1}{\tau}\int_o^\infty\frac{d}{dt}\left(\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right)dt - \langle\sum_\alpha\mathbf{F}_\alpha\cdot\mathbf{r}_\alpha\rangle \\ &= \lim_{\tau\to\infty}\frac{1}{\tau}\left[\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right]_0^\infty + \langle\sum_\alpha\frac{\partial U}{\partial\mathbf{r}_\alpha}\cdot\mathbf{r}_\alpha\rangle \\ &= k\langle U \rangle \end{aligned}\]

Hamiltonian Mechanics / Hamilton力学

Hamilton’s Equation / Hamilton方程式

\[H = \sum_i p_i \dot{q}_i - L \quad p_i = \frac{\partial{L}}{\partial{\dot{q}_i}} \\ \dot{q}_i = \frac{\partial{H}}{\partial{p}_i} \quad \dot{p}_i = -\frac{\partial{H}}{\partial{q}_i}\]

\[\begin{aligned}{} dH &= \sum_i \left(\dot{q}_i dp_i + p_i d\dot{q}_i \right) - \sum_i \left(\frac{\partial{L}}{\partial{q_i}} dq_i + \frac{\partial{L}}{\partial{\dot{q}_i}}d\dot{q_i}\right) - \frac{\partial{L}}{\partial{t}}dt \\ &= \sum_i \left(\dot{q}_i dp_i - \dot{p}_i dq_i\right) - \frac{\partial{L}}{\partial{t}} dt \end{aligned} \\ \dot{q}_i = \frac{\partial{H}}{\partial{p_i}} \quad \dot{p}_i = -\frac{\partial{H}}{\partial{q_i}} \quad \frac{\partial{L}}{\partial{t}} = -\frac{\partial{H}}{\partial{t}}\]

Poisson Bracket / Poisson括弧

\[\{F, G\} = \sum_i \left(\frac{\partial{F}}{\partial{q_i}} \frac{\partial{G}}{\partial{p_i}} - \frac{\partial{F}}{\partial{p_i}} \frac{\partial{G}}{\partial{q_i}}\right) \\ \frac{dF}{dt} = \{F, H\} + \frac{\partial{F}}{\partial{t}}\]

\[\begin{aligned}{} \frac{dF}{dt} &= \sum_i \left(\frac{\partial{F}}{\partial{q_i}}\dot{q}_i + \frac{\partial{F}}{\partial{p_i}}\dot{p}_i \right) + \frac{\partial{F}}{\partial{t}} \\ &= \sum_i \left(\frac{\partial{F}}{\partial{q_i}} \frac{\partial{H}}{\partial{p_i}} - \frac{\partial{F}}{\partial{p_i}} \frac{\partial{H}}{\partial{q_i}}\right) + \frac{\partial{F}}{\partial{t}} \\ &= \{F, H\} + \frac{\partial{F}}{\partial{t}} \end{aligned}\]

Canonical Transformation / 正準変換

\[\begin{aligned}{} & W(q,Q,t): && p = \frac{\partial{W}}{\partial{q}} && P = -\frac{\partial{W}}{\partial{Q}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(q,P,t): && p = \frac{\partial{W}}{\partial{q}} && Q=\frac{\partial{W}}{\partial{P}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(p,Q,t): && q = -\frac{\partial{W}}{\partial{p}} && P = -\frac{\partial{W}}{\partial{Q}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(p,P,t): && q = -\frac{\partial{W}}{\partial{p}} && Q = \frac{\partial{W}}{\partial{P}} && H' = H + \frac{\partial{W}}{\partial{t}} \end{aligned}\]

\[\delta \int_{t_1}^{t_2} \left(\sum_i p_i \dot{q}_i - H\right) dt = 0 \quad \delta \int_{t_1}^{t_2} \left(\sum_i P_i \dot{Q}_i - H'\right) dt = 0 \\ \sum_i p_i dq_i - H dt = \sum_i P_i dQ_i - H' dt + dW_1(q,Q,t) \\ dW_1 = \sum_i p_i dq_i - \sum_i P_i dQ_i + (H' - H) dt \\ p_i = \frac{\partial{W_1}}{\partial{q_i}} \quad P_i = -\frac{\partial{W_1}}{\partial{Q_i}} \quad H' = H + \frac{\partial{W_1}}{\partial{t}} \\ W_2 = W_1 + \sum_i P_i Q_i \\ dW_2 = \sum_i p_i dq_i + \sum_i Q_i dP_i + (H' - H) dt \\ p_i = \frac{\partial{W_2}}{\partial{q_i}} \quad Q_i = \frac{\partial{W_2}}{\partial{P_i}} \quad H' = H + \frac{\partial{W_2}}{\partial{t}} \\ W_3 = W_1 - \sum_i p_i q_i \\ dW_3 = -\sum_i q_i dp_i - \sum_i P_i dQ_i + (H' - H) dt \\ q_i = -\frac{\partial{W_3}}{\partial{p_i}} \quad P_i = -\frac{\partial{W_3}}{\partial{Q_i}} \quad H' = H + \frac{\partial{W_3}}{\partial{t}} \\ W_4 = W_1 - \sum_i p_i q_i + \sum_i P_i Q_i \\ dW_4 = - \sum_i q_i dp_i + \sum_i Q_i dP_i + (H' - H) dt \\ q_i = -\frac{\partial{W_4}}{\partial{p_i}} \quad Q_i = \frac{\partial{W_4}}{\partial{P_i}} \quad H' = H + \frac{\partial{W_4}}{\partial{t}}\]

Hamilton–Jacobi Equation / Hamilton–Jacobi方程式

\[\frac{\partial{S}}{\partial{t}} + H\left(q,\frac{\partial{S}}{\partial{q}},t\right) = 0\]

\[H' = H + \frac{\partial{W}}{\partial{t}} = 0 \\ \dot{P}_i = -\frac{\partial{H'}}{\partial{Q_i}} = 0 \quad P_i = \alpha_i \\ \begin{aligned}{} \frac{dW}{dt} &= \sum_i \frac{\partial{W}}{\partial{q_i}} \dot{q}_i + \frac{\partial{W}}{\partial{t}} \\ &= \sum_i p_i \dot{q}_i - H \\ &= L \end{aligned} \\ W = \int L dt = S\]

Liouville’s Theorem / Liouvilleの定理

\[\int_R \prod_i dq_i dp_i = \int_{R'} \prod_i dQ_i dP_i\]

\[\prod_i dQ_i dP_i = \frac{\partial{(Q,P)}}{\partial{(q,p)}} \prod_i dq_i dp_i \\ \begin{aligned}{} \frac{\partial{(Q,P)}}{\partial{(q,p)}} &= \frac{\partial{(Q,P)}}{\partial{(q,P)}} \frac{\partial{(q,P)}}{\partial{(q,p)}} = \frac{\partial{(Q)}}{\partial{(q)}} \frac{\partial{(P)}}{\partial{(p)}} \\ &= \frac{\frac{\partial{(Q)}}{\partial{(q)}}}{\frac{\partial{(p)}}{\partial{(P)}}} = \frac{\det\left(\frac{\partial^2 W}{\partial q_j \partial P_i}\right)}{\det\left(\frac{\partial^2 W}{\partial P_j \partial q_i}\right)} \\ &= 1 \end{aligned}\]

Oscillation / 振動

Small Oscillation / 微小振動

\[\ddot{x} + \omega_0^2 x = 0 \\ x = \tilde{A} e^{i \omega_0 t}\]

\[U'(x_0) = 0 \quad U''(x_0) = k \\ L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2 \\ m \ddot{x} + k x = 0 \\ \ddot{x} + \omega_0^2 x = 0 \\ x = \tilde{A} e^{i \omega_0 t}\]

Damped Oscillation / 減衰振動

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0 \\ \begin{aligned}{} x &= e^{-\beta t} \cdot \tilde{A} e^{i \sqrt{\omega_0^2 - \beta^2} t} \\ x &= e^{-\beta t} (A + B t) \\ x &= e^{-\beta t} \left( A_1 e^{\sqrt{\beta^2 - \omega_0^2} t} + A_2 e^{-\sqrt{\beta^2 - \omega_0^2} t} \right) \end{aligned}\]

\[m \ddot{x} + b \dot{x} + k x = 0 \\ \ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0 \\ x = e^{\gamma t} \\ \gamma^2 + 2 \beta \gamma + \omega_0^2 = 0 \\ \gamma = -\beta \pm \sqrt{\beta^2 - \omega_0^2} \\ x = e^{-\beta t} \left( A_1 e^{\sqrt{\beta^2 - \omega_0^2} t} + A_2 e^{-\sqrt{\beta^2 - \omega_0^2} t} \right)\]

Forced Oscillation / 強制振動

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = \alpha e^{i \omega t} \\ x_p = \tilde{A} e^{i \omega t} \\ |\tilde{A}| = \frac{\alpha}{\sqrt{(\omega^2 - \omega_0^2)^2 + 4 \beta^2 \omega^2}} \\ \text{arg}(\tilde{A}) = \tan^{-1} \left( \frac{2 \beta \omega}{\omega^2 - \omega_0^2} \right)\]

\[m \ddot{x} + b \dot{x} + k x = f\cos(\omega t) \\ \ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = \alpha e^{i \omega t} \\ x_p = \tilde{A} e^{i \omega t} \\ (-\omega^2 + 2 \beta \omega i + \omega_0^2) \tilde{A} = \alpha \\ \tilde{A} = \frac{\alpha}{(\omega_0^2 - \omega^2) + 2 \beta \omega i}\]

Coupled Oscillation / 連成振動

\[M \ddot{\mathbf{x}} + K \mathbf{x} = 0 \\ \left( -\omega_n^2 M + K \right) \mathbf{A_n} = 0 \\ \mathbf{x} = \sum_n \tilde{c}_n \mathbf{A_n} e^{i \omega_n t} \\ \mathbf{x} = \sum_n q_n \mathbf{A_n} \quad \ddot{q}_n + \omega_n^2 q_n = 0\]

\[\begin{aligned}{} L &= \frac{1}{2} \sum_{i,j} m_{ij} \dot{x}_i \dot{x}_j - \frac{1}{2} \sum_{i,j} k_{ij} x_i x_j \\ &= \frac{1}{2} \dot{\mathbf{x}}^T M \dot{\mathbf{x}} - \frac{1}{2} \mathbf{x}^T K \mathbf{x} \end{aligned} \\ M \ddot{\mathbf{x}} + K \mathbf{x} = 0 \quad \mathbf{x} = \mathbf{A} e^{i \omega t} \\ |-\omega_n^2 M + K| = 0 \quad (-\omega_n^2 M + K) \mathbf{A}_n = 0 \\ \mathbf{x} = \sum_n \hat{c}_n \mathbf{A}_n e^{i \omega_n t} \\ \mathbf{A}_n^T (K \mathbf{A}_m) - (K \mathbf{A}_n)^T \mathbf{A}_m = (\omega_n^2 - \omega_m^2) \mathbf{A}_n^T M \mathbf{A}_m = 0 \\ \mathbf{A}_n^T M \mathbf{A}_m = \delta_{nm} \quad \mathbf{A}_n^T K \mathbf{A}_m = \delta_{nm} \omega_m^2 \\ \mathbf{x} = \sum_n q_n \mathbf{A}_n \\ \begin{aligned}{} L &= \frac{1}{2} \sum_{n,m} \dot{q}_n (\mathbf{A}_n^T M \mathbf{A}_m) \dot{q}_m - \frac{1}{2} \sum_{n,m} q_n (\mathbf{A}_n^T K \mathbf{A}_m) q_m \\ &= \frac{1}{2} \sum_n \dot{q}_n^2 - \frac{1}{2} \sum_n \omega_n^2 q_n^2 \end{aligned} \\ \ddot{q}_n + \omega_n^2 q_n = 0\]

Wave / 波動

Wave Equation / 波動方程式

\[\frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) + \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) - \frac{\delta L}{\delta \phi} = 0 \\ \rho \frac{\partial^2 \phi}{\partial t^2} - k \frac{\partial^2 \phi}{\partial x^2} = 0\]

\[L = \int \mathcal{L} \left( \phi, \frac{\partial \phi}{\partial t}, \frac{\partial \phi}{\partial x}, t \right) dx \\ \begin{aligned}{} \delta S &= \int dt \int dx \left[ \frac{\delta L}{\delta \phi} \delta \phi + \frac{\delta L}{\delta (\partial_t \phi)} \delta (\partial_t \phi) + \frac{\delta L}{\delta (\partial_x \phi)} \delta (\partial_x \phi) \right] \\ &= \int dt \int dx \left[ \frac{\delta L}{\delta \phi} - \frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) - \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) \right] \delta \phi \end{aligned} \\ \frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) + \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) - \frac{\delta L}{\delta \phi} = 0 \\ L = \int \left[ \frac{1}{2} \rho (\partial_t \phi)^2 - \frac{1}{2} k (\partial_x \phi)^2 \right] dx \\ \delta L = \int \left[ \rho (\partial_t \phi) \delta (\partial_t \phi) - k (\partial_x \phi) \delta (\partial_x \phi) \right] dx \\ \frac{\delta L}{\delta (\partial_t \phi)} = \rho \frac{\partial \phi}{\partial t} \quad \frac{\delta L}{\delta (\partial_x \phi)} = -k \frac{\partial \phi}{\partial x} \\ \rho \frac{\partial^2 \phi}{\partial t^2} - k \frac{\partial^2 \phi}{\partial x^2} = 0\]

Bounded Wave / 有界波

\[\frac{\partial^2 \phi}{\partial t^2} - v^2 \frac{\partial^2 \phi}{\partial x^2} = 0 \\ \phi = \sum_n \tilde{A}_n e^{i (\omega_n t - k_n x)} \\ \begin{aligned}{} &\phi(0) = \phi(L) = 0: && k_n = \frac{n \pi}{L} \quad \tilde{A}_{-n} = -\tilde{A}_n \\ &\frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) = 0: && k_n = \frac{n \pi}{L} \quad \tilde{A}_{-n} = \tilde{A}_n \\ &\phi(0) = \phi(L),\; \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L): && k_n = \frac{2n \pi}{L} \end{aligned}\]

\[\phi(x,t) = \psi(x) e^{i \omega t} \\ \psi(x) = \tilde{A} e^{-ikx} \\ \phi = \sum_n \tilde{A}_n e^{i(\omega_n t - k_n x)} \quad k_n = \pm \frac{\omega_n}{v} \\ \phi(0) = \phi(L) = 0 \\ \begin{aligned}{} &\tilde{A}_n e^{i \omega_n t} + \tilde{A}_{-n} e^{i \omega_n t} = 0 && \tilde{A}_{-n} = -\tilde{A}_n \\ &\tilde{A}_n e^{i \omega_n t} \left( e^{-ik_n L} - e^{ik_n L} \right) = 0 && k_n L = n \pi \end{aligned} \\ \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) = 0 \\ \begin{aligned}{} & -\tilde{A}_n k_n e^{i \omega_n t} - \tilde{A}_{-n} (-k_n) e^{i \omega_n t} = 0 && \tilde{A}_{-n} = \tilde{A}_n \\ & -\tilde{A}_n k_n e^{i \omega_n t} \left( e^{-ik_n L} - e^{ik_n L} \right) = 0 && k_n L = n \pi \end{aligned} \\ \phi(0) = \phi(L) \quad \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) \\ \begin{align*} \left( \tilde{A}_n + \tilde{A}_{-n} \right) e^{i\omega_n t} &= \left(\tilde{A}_n e^{-i k_n L} + \tilde{A}_{-n} e^{i k_n L} \right) e^{i\omega_n t} \\ \left( -\tilde{A}_n k_n + \tilde{A}_{-n} k_n \right) e^{i\omega_n t} &= \left( -\tilde{A}_n k_n e^{-i k_n L} + \tilde{A}_{-n} k_n e^{i k_n L} \right) e^{i\omega_n t} \end{align*} \\ e^{ik_n L} = e^{-ik_n L} = 1 \quad k_n L = 2n \pi\]

Free Wave / 自由波

\[\frac{\partial^2 \phi}{\partial t^2} - v^2 \frac{\partial^2 \phi}{\partial x^2} = 0 \\ \phi = \int \tilde{A}(k) e^{i (\omega t - k x)} \, dk \\ v_p = \frac{\omega}{k} \quad v_g = \frac{d\omega}{dk}\]

\[\phi(x, t) = \psi(x) e^{i \omega t} \\ \psi(x) = \tilde{A} e^{-i k x} \\ \phi = \int \tilde{A}(k) e^{i (\omega t - k x)} \, dk \quad k = \frac{\omega}{v} \\ \omega = \omega_0 + \left( \frac{d\omega}{dk} \right)_0 (k - k_0) \\ \omega t - k x = (\omega_0 t - k_0 x) + (k - k_0)(\omega_0' t - x) \\ \phi = \int \tilde{A}(k) e^{i(\omega_0 t - k_0 x)} e^{i (k - k_0) (\omega_0' t - x)} \, dk \\ v_p = \frac{\omega}{k} \quad v_g = \frac{d\omega}{dk}\]

Central Force Motion / 中心力運動

Reduced Mass / 換算質量

\[\begin{aligned}{} \mu &= \frac{m_1 m_2}{m_1 + m_2} \\ M &= \mu r^2 \dot{\theta} \\ E &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2}\frac{M^2}{\mu r^2} + U(r) \end{aligned}\]

\[\begin{aligned}{} L &= \frac{1}{2} m_1 \left| \frac{m_2}{m_1 + m_2} \dot{\mathbf{r}} \right|^2 + \frac{1}{2} m_2 \left| \frac{m_1}{m_1 + m_2} \dot{\mathbf{r}} \right|^2 - U(r) \\ &= \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\dot{r}^2 + r^2 \dot{\theta}^2) - U(r) \end{aligned} \\ M = \frac{\partial L}{\partial\dot{\theta}} = \mu r^2 \dot{\theta} \\ \begin{aligned}{} E &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2} \mu r^2 \dot{\theta}^2 + U(r) \\ &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2}\frac{M^2}{\mu r^2} + U(r) \end{aligned}\]

Orbit / 軌道

\[U_{\text{eff}}(r) = U(r) + \frac{M^2}{2\mu r^2} \\ t = \int \frac{dr}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \\ \theta = \int \frac{M dr}{r^2 \sqrt{2\mu (E - U_{\text{eff}}(r))}}\]

\[E = \frac{1}{2} \mu \dot{r}^2 + U_{\text{eff}}(r) \\ \frac{dr}{dt} = \sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))} \\ t = \int \frac{dr}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \\ \begin{aligned}{} \frac{d\theta}{dr} &= \frac{d\theta}{dt} \frac{dt}{dr} \\ &= \frac{M}{\mu r^2} \frac{1}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \end{aligned} \\ \theta = \int \frac{M dr}{r^2 \sqrt{2\mu (E - U_{\text{eff}}(r))}}\]

Kepler Problem / Kepler問題

\[U = -\frac{\alpha}{r} \quad p = \frac{M^2}{\mu\alpha} \quad e = \sqrt{1 + \frac{2EM^2}{\mu \alpha^2}} \\ r = \frac{p}{1 + e \cos \theta} \\ \tau = 2\pi \sqrt{\frac{\mu}{\alpha}}a^{3/2}\]

\[\begin{aligned}{} \theta &= \int \frac{\frac{M}{r^2} \, dr}{\sqrt{2\mu \left( E + \frac{\alpha}{r} - \frac{M^2}{2\mu r^2} \right)}} \\ &= \int \frac{\frac{1}{r^2} \,dr}{\sqrt{ -\left( \frac{1}{r} - \frac{\mu \alpha}{M^2}\right)^2 + \frac{\mu^2 \alpha^2}{M^4}\left(1 + \frac{2EM^2}{\mu \alpha^2}\right)}} \\ &= \int \frac{\frac{1}{r^2} \,dr}{\sqrt{ -\left( \frac{1}{r} - \frac{1}{p}\right)^2 + \frac{e^2}{p^2}}} \\ &= \int \frac{\frac{e}{p}\sin\phi\,d\phi}{\frac{e}{p}\sin \phi} \\ &= \phi + C \quad \left( \frac{1}{r} - \frac{1}{p} = \frac{e}{p} \cos \phi \right) \end{aligned} \\ r = \frac{p}{1 + e \cos \theta} \quad a = \frac{p}{1 - e^2} \quad b = \frac{p}{\sqrt{1 - e^2}} \\ \tau = \frac{\pi ab}{\dot{S}} = \frac{\pi a \sqrt{ap}}{\frac{M}{2\mu}}= 2\pi \sqrt{\frac{\mu}{\alpha}}a^{3/2}\]

Collision / 衝突

\[\begin{aligned}{} \mathbf{v}_1 &= \frac{m_2 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \\ \mathbf{v}_2 &= \frac{-m_1 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \end{aligned} \\ \begin{aligned}{} \tan{\theta_1} &= \frac{\sin{\theta}}{\cos{\theta} + \frac{m_1}{m_2}} \\ \tan{\theta_2} &= \frac{\sin{\theta}}{\cos{\theta} - 1} \end{aligned}\]

\[m_1 \mathbf{v}_1' + m_2 \mathbf{v}_2' = 0 \\ \mathbf{v}_1' - \mathbf{v}_2' = \mathbf{v} \\ \begin{aligned}{} \mathbf{v}_1 &= \mathbf{v}_1' + \mathbf{V} = \frac{m_2 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \\ \mathbf{v}_2 &= \mathbf{v}_2' + \mathbf{V} = \frac{-m_1 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \end{aligned} \\ u_2 = 0 \quad u_1 = v \\ \begin{aligned}{} \tan{\theta_1} &= \frac{m_2 \mathbf{v} \sin{\theta}}{m_2 \mathbf{v} \cos{\theta} + m_1 \mathbf{v}} = \frac{\sin{\theta}}{\cos{\theta} + \frac{m_1}{m_2}} \\ \tan{\theta_2} &= \frac{-m_1 \mathbf{v} \sin{\theta}}{-m_1 \mathbf{v} \cos{\theta} + m_1 \mathbf{v}} = \frac{\sin{\theta}}{\cos{\theta} - 1} \end{aligned}\]

Scattering / 散乱

\[\begin{aligned}{} \frac{d\sigma}{d\Omega} &= \frac{R^2}{4} \\ \frac{d\sigma}{d\Omega} &= \left(\frac{\alpha}{4T}\right)^2 \frac{1}{\sin^4\frac{\theta}{2}} \end{aligned}\]

\[b = R \sin{\varphi} = R \sin{\left(\frac{\pi}{2} - \frac{\theta}{2}\right)} = R \cos{\frac{\theta}{2}} \\ \begin{aligned}{} \frac{d\sigma}{d\Omega} &= \frac{b}{\sin{\theta}} \left| \frac{db}{d\theta} \right| \\ &= \frac{R \cos{\frac{\theta}{2}}}{2 \sin{\frac{\theta}{2}} \cos{\frac{\theta}{2}}} \cdot \frac{R}{2} \sin{\frac{\theta}{2}} \\ &= \frac{R^2}{4} \end{aligned} \\ \frac{1}{r_\infty} + \frac{1}{p} = \frac{e}{p} \cos{\varphi} \quad \quad \cos{\varphi} = \frac{1}{e} \\ e = \sqrt{1 + \frac{2EM^2}{m \alpha^2}} = \sqrt{1 + \frac{4T^2 b^2}{\alpha^2}} \\ \tan{\varphi} = \sqrt{e^2 - 1} = \frac{2Tb}{\alpha} \\ b = \frac{\alpha}{2T} \tan{\varphi} = \frac{\alpha}{2T} \tan{\left(\frac{\pi}{2} - \frac{\theta}{2}\right)} = \frac{\alpha}{2T} \cot{\frac{\theta}{2}} \\ \begin{aligned}{} \frac{d\sigma}{d\Omega} &= \frac{b}{\sin{\theta}} \left| \frac{db}{d\theta} \right| \\ &= \frac{\frac{\alpha}{2T} \cot{\frac{\theta}{2}}}{2 \sin{\frac{\theta}{2}} \cos{\frac{\theta}{2}}} \cdot \frac{\alpha}{4T} \frac{1}{\sin^2{\frac{\theta}{2}}} \\ &= \left(\frac{\alpha}{4T}\right)^2 \frac{1}{\sin^4{\frac{\theta}{2}}} \end{aligned}\]

Gravity / 重力

\[\begin{aligned}{} \Phi &= \frac{U}{m} = -\frac{GM}{r} \\ \mathbf{g} &= -\nabla \Phi = -\frac{GM}{r^2} \hat{\mathbf{e}}_r \\ \mathbf{F} &= m\mathbf{g} = -\frac{GMm}{r^2} \hat{\mathbf{e}}_r \end{aligned}\]

Rigid Body Motion / 剛体運動

Non-Inertial Frame / 非慣性系

\[\begin{aligned}{} \mathbf{r} &= \mathbf{R} + \mathbf{r}' \\ \mathbf{v} &= \mathbf{V} + \mathbf{v}' + \boldsymbol{\Omega} \times \mathbf{r}' \\ \mathbf{a} &= \mathbf{A} + \mathbf{a}' + \dot{\boldsymbol{\Omega}} \times \mathbf{r}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}') + 2 \boldsymbol{\Omega} \times \mathbf{v}' \end{aligned}\]

\[\frac{d\mathbf{Q}}{dt} = \frac{d'\mathbf{Q}}{dt} + \boldsymbol{\Omega} \times \mathbf{Q} \\ \mathbf{r} = \mathbf{R} + \mathbf{r}' \\ \begin{aligned}{} \frac{d\mathbf{r}}{dt} &= \frac{d\mathbf{R}}{dt} + \frac{d\mathbf{r}'}{dt} \\ &= \frac{d\mathbf{R}}{dt} + \frac{d'\mathbf{r}'}{dt} + \boldsymbol{\Omega} \times \mathbf{r}' \end{aligned} \\ \mathbf{v} = \mathbf{V} + \mathbf{v}' + \boldsymbol{\Omega} \times \mathbf{r}' \\ \begin{aligned}{} \frac{d\mathbf{v}}{dt} &= \frac{d\mathbf{V}}{dt} + \frac{d\mathbf{v}'}{dt} + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{r}' + \boldsymbol{\Omega} \times \frac{d\mathbf{r}'}{dt} \\ &= \frac{d\mathbf{V}}{dt} + \frac{d'\mathbf{v}'}{dt} + \boldsymbol{\Omega}\times\mathbf{v}' + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{r}' + \boldsymbol{\Omega} \times \left( \frac{d'\mathbf{r}'}{dt} + \boldsymbol{\Omega} \times \mathbf{r}' \right) \end{aligned} \\ \mathbf{a} = \mathbf{A} + \mathbf{a}' + \dot{\boldsymbol{\Omega}} \times \mathbf{r}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}') + 2 \boldsymbol{\Omega} \times \mathbf{v}'\]

Euler Angle / Euler角

\[\begin{aligned}{} \Omega_1 &= \dot{\phi} \sin \theta \sin \psi + \dot{\theta} \cos \psi \\ \Omega_2 &= \dot{\phi} \sin \theta \cos \psi - \dot{\theta} \sin \psi \\ \Omega_3 &= \dot{\phi} \cos \theta + \dot{\psi} \end{aligned} \\\]

Inertia Tensor / 慣性テンソル

\[\begin{aligned}{} I_{ij} &= \sum_{\alpha} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \\ T &= \sum_{i,j} \frac{1}{2} I_{ij} \Omega_i \Omega_j \\ M_i &= \sum_j I_{ij} \Omega_j \end{aligned}\]

\[\begin{aligned}{} T &= \sum_{\alpha} \frac{1}{2} m_{\alpha} (\boldsymbol{\Omega} \times \mathbf{r}_{\alpha}) \cdot (\boldsymbol{\Omega} \times \mathbf{r}_{\alpha}) \\ &= \sum_{\alpha} \frac{1}{2} m_{\alpha} [\Omega^2 r_{\alpha}^2 - (\boldsymbol{\Omega}\cdot\mathbf{r}_{\alpha})^2] \\ &= \sum_{i,j}\sum_{\alpha} \frac{1}{2} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \Omega_i \Omega_j \\ \end{aligned} \\ \begin{aligned}{} \mathbf{M} &= \sum_{\alpha} m_{\alpha} \mathbf{r}_{\alpha} \times (\mathbf{\Omega} \times \mathbf{r}_{\alpha}) \\ &= \sum_{\alpha} m_{\alpha} [\mathbf{\Omega} (r_{\alpha}^2) - \mathbf{r}_{\alpha} (\mathbf{r}_{\alpha} \cdot \Omega)] \\ M_i &= \sum_j \sum_{\alpha} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \Omega_j \end{aligned}\]

Euler’s Equation / Eulerの運動方程式

\[\begin{aligned}{} I_1 \dot{\Omega}_1 + (I_3 - I_2) \Omega_2 \Omega_3 &= N_1 \\ I_2 \dot{\Omega}_2 + (I_1 - I_3) \Omega_3 \Omega_1 &= N_2 \\ I_3 \dot{\Omega}_3 + (I_2 - I_1) \Omega_1 \Omega_2 &= N_3 \end{aligned}\]

\[\frac{d\mathbf{M}}{dt} = \frac{d'\mathbf{M}}{dt} + \boldsymbol{\Omega} \times \mathbf{M} = \mathbf{N} \\ \begin{bmatrix} I_1 \dot{\Omega}_1 \\ I_2 \dot{\Omega}_2 \\ I_3 \dot{\Omega}_3 \end{bmatrix} + \begin{bmatrix} \Omega_1 \\ \Omega_2 \\ \Omega_3 \end{bmatrix} \times \begin{bmatrix} I_1 \Omega_1 \\ I_2 \Omega_2 \\ I_3 \Omega_3 \end{bmatrix} = \begin{bmatrix} N_1 \\ N_2 \\ N_3 \end{bmatrix}\]

Symmetrical Top (Free) / 対称コマ（自由）

\[\begin{aligned}{} \theta &= \text{const.} \\ \dot{\phi} &= \frac{M_z}{I_1} \\ \dot{\psi} &= M_z \cos \theta \left( \frac{1}{I_3} - \frac{1}{I_1} \right) \end{aligned}\]

\[L = \frac{1}{2} I_1 (\dot{\phi}^2 \sin^2 \theta + \dot{\theta}^2) + \frac{1}{2} I_3 (\dot{\phi} \cos \theta + \dot{\psi})^2 \\ \begin{aligned}{} \frac{\partial L}{\partial \dot{\psi}} &= I_3 (\dot{\phi} \cos \theta + \dot{\psi}) = M_3 = M_z \cos \theta \\ \frac{\partial L}{\partial \dot{\phi}} &= I_1 \dot{\phi} \sin^2 \theta + I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \cos \theta = M_z \end{aligned} \\ \begin{aligned}{} \dot{\phi} &= \frac{M_z - M_z \cos^2 \theta}{I_1 \sin^2 \theta} = \frac{M_z}{I_1} \\ \dot{\psi} &= \frac{M_z \cos \theta}{I_3} - \dot{\phi} \cos \theta = M_z \cos \theta \left( \frac{1}{I_3} - \frac{1}{I_1} \right) \end{aligned} \\ \begin{aligned}{} \frac{\partial L}{\partial \theta} &= I_1 \dot{\phi}^2 \sin \theta \cos \theta - I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \dot{\phi} \sin \theta \\ &= \frac{M_z^2}{I_1} \sin \theta \cos \theta - \frac{M_z^2}{I_1} \sin \theta \cos \theta = 0 \\ &= \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) = \frac{d}{dt} (I_1 \dot{\theta}) \quad (\dot{\theta} = \text{const.}) \end{aligned} \\ E = \frac{1}{2} I_1 \left( \frac{M_z^2}{I_1^2} \sin^2 \theta + \dot{\theta}^2 \right) + \frac{1}{2} I_3 \frac{M_z^2}{I_3^2} \cos^2 \theta \\ \quad \theta = \text{const.}\]

Symmetrical Top (Gravity) / 対称コマ（重力）

\[\begin{aligned}{} \theta_1 &\leq \theta \leq \theta_2 \\ \dot{\phi}& = \frac{M_z - M_3 \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \\ \dot{\psi} &= \frac{M_3}{I_3} - \frac{(M_z - M_3 \cos \theta) \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \end{aligned}\]

\[L = \frac{1}{2} (I_1 + Ma^2) (\dot{\phi}^2 \sin^2 \theta + \dot{\theta}^2) + \frac{1}{2} I_3 (\dot{\phi} \cos \theta + \dot{\psi})^2 - Mga \cos \theta \\ \begin{aligned}{} \frac{\partial L}{\partial \dot{\psi}} &= I_3 (\dot{\phi} \cos \theta + \dot{\psi}) = M_3 \\ \frac{\partial L}{\partial \dot{\phi}} &= (I_1 + Ma^2) \dot{\phi} \sin^2 \theta + I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \cos \theta = M_z \end{aligned} \\ \begin{aligned}{} \dot{\phi} &= \frac{M_z - M_3 \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \\ \dot{\psi} &= \frac{M_3}{I_3} - \frac{(M_z - M_3 \cos \theta) \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \end{aligned} \\ E = \frac{1}{2} (I_1 + Ma^2) \dot{\theta}^2 + \frac{1}{2} \frac{(M_z - M_3 \cos \theta)^2}{(I_1 + Ma^2) \sin^2 \theta} + \frac{1}{2} \frac{M_3^2}{I_3} + Mga \cos \theta \\ u = \cos \theta \quad \dot{\theta}^2 = \frac{\dot{u}^2}{1 - u^2} \\ E = \frac{1}{2} (I_1 + Ma^2) \frac{\dot{u}^2}{1 - u^2} + \frac{1}{2} \frac{(M_z - M_3 u)^2}{(I_1 + Ma^2)(1 - u^2)} + \frac{1}{2} \frac{M_3^2}{I_3} + Mga u \\ \dot{u}^2 = (1 - u^2) \left( A - B u \right) - \frac{(M_z - M_3 u)^2}{(I_1 + Ma^2)^2} \geq 0 \\ f(\cos \theta_1) = f(\cos \theta_2) = 0 \\ \theta_1 \leq \theta \leq \theta_2\]

Asymmetrical Top (Free) / 非対称コマ（自由）

\[\sqrt{2EI_1} \leq M \leq \sqrt{2EI_3} \\ \Omega_1: \text{stable} \quad \Omega_2: \text{unstable} \quad \Omega_3: \text{stable}\]

\[M^2 = M_1^2 + M_2^2 + M_3^2 \\ E = \frac{M_1^2}{2I_1} + \frac{M_2^2}{2I_2} + \frac{M_3^2}{2I_3} \\ \sqrt{2EI_1} \leq M \leq \sqrt{2EI_3} \\ \delta \Omega_1 = a e^{\lambda t} \quad \delta \Omega_2 = b e^{\lambda t} \quad \delta \Omega_3 = c e^{\lambda t} \\ \begin{aligned}{} \Omega_1 : \quad \lambda^2 &= \frac{-\Omega_1^2(I_3 - I_1)(I_2 - I_1)}{I_2 I_3} < 0, &&\text{stable} \\ \Omega_2 : \quad \lambda^2 &= \frac{\Omega_2^2(I_3 - I_2)(I_2 - I_1)}{I_1 I_3} > 0, &&\text{unstable} \\ \Omega_3 : \quad \lambda^2 &= \frac{-\Omega_3^2(I_3 - I_2)(I_3 - I_1)}{I_1 I_2} < 0, &&\text{stable} \end{aligned}\]

Continuum Mechanics / 連続体力学

Fundamental Equation / 基礎方程式

\[\frac{DQ}{Dt} = \frac{\partial Q}{\partial t} + \mathbf{v} \cdot \nabla Q \\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + \nabla \cdot \sigma\]

\[\frac{d}{dt} \int_V \rho \, dV = \int_V \frac{\partial \rho}{\partial t} \, dV = -\oint_{\partial V} \rho \mathbf{v} \cdot d\mathbf{S} = -\int_V \nabla \cdot (\rho \mathbf{v}) \, dV \\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \\ \begin{aligned}{} \frac{d}{dt} \int_V \rho v_i \, dV &= \int_V \frac{\partial (\rho v_i)}{\partial t} \, dV \\ &= - \oint_{\partial V} \rho v_i \mathbf{v} \cdot d\mathbf{S} + \int_V f_i \, dV + \oint_{\partial V} \boldsymbol{\sigma}_i \cdot d\mathbf{S} \\ &= - \int_V \nabla \cdot (\rho v_i \mathbf{v}) \, dV + \int_V f_i \, dV + \int_V \nabla \cdot \boldsymbol{\sigma}_i \, dV \end{aligned} \\ \begin{aligned}{} \frac{\partial (\rho v_i)}{\partial t} + \nabla \cdot (\rho v_i \mathbf{v}) &= \frac{\partial \rho}{\partial t} v_i + \rho \frac{\partial v_i}{\partial t} + (\nabla \cdot \rho \mathbf{v}) v_i + \rho (\mathbf{v} \cdot \nabla) v_i \\ &= \rho \left( \frac{\partial v_i}{\partial t} + (\mathbf{v} \cdot \nabla) v_i \right) + \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) \right) v_i \\ &= f_i + \nabla \cdot \boldsymbol{\sigma}_i \end{aligned} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + \nabla \cdot \sigma\]

Elastic Body / 弾性体

\[E_{ij} = \frac{1}{2} \left( \frac{\partial r_j}{\partial x_i} + \frac{\partial r_i}{\partial x_k} \right) \\ {\sigma}_{ij} = \lambda \delta_{ij} E_{kk} + 2 \mu E_{ij} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{r}) + \mu \nabla^2 \mathbf{r}\]

\[\begin{aligned}{} \rho \frac{Dv_i}{Dt} &= f_i + \frac{\partial {\sigma}_{ij}}{\partial x_j} \\ &= f_i + \lambda \frac{\partial}{\partial x_i} \left( \frac{\partial r_k}{\partial x_k} \right) + \mu \frac{\partial}{\partial x_j} \left( \frac{\partial r_j}{\partial x_i} + \frac{\partial r_i}{\partial x_j} \right) \\ &= f_i + (\lambda + \mu) \frac{\partial}{\partial x_i} (\nabla \cdot \mathbf{r}) + \mu \nabla^2 r_i \end{aligned}\]

Fluid / 流体

\[\dot{E}_{ij} = \frac{1}{2} \left( \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right) \\ {\sigma}_{ij} = - p \delta_{ij} + \lambda \delta_{ij} \dot{E}_{kk} + 2 \eta \dot{E}_{ij} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} - \nabla p + (\lambda + \eta) \nabla (\nabla \cdot \mathbf{v}) + \eta {\nabla}^2 \mathbf{v}\]

\[\begin{aligned}{} \rho \frac{D v_i}{Dt} &= f_i + \frac{\partial {\sigma}_{ij}}{\partial x_j} \\ &= f_i - \frac{\partial p}{\partial x_i} + \lambda \frac{\partial}{\partial x_i} \left( \frac{\partial v_k}{\partial x_k} \right) + \eta \frac{\partial}{\partial x_j} \left( \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right) \\ &= f_i - \frac{\partial p}{\partial x_i} + (\lambda + \eta) \frac{\partial}{\partial x_i} (\nabla \cdot \mathbf{v}) + \eta {\nabla}^2 v_i \end{aligned}\]

Bernoulli’s Principle / Bernoulliの定理

\[\rho \mathbf{v} \cdot \mathbf{s} = \text{const.} \\ p + \frac{1}{2} \rho v^2 + \rho gz = \text{const.}\]

\[\lambda = \eta = 0 \quad \frac{\partial \rho}{\partial t} = \frac{\partial \mathbf{v}}{\partial t} = 0 \\ \begin{aligned}{} \int_{V} \frac{\partial \rho}{\partial t} \, dV &= \int_{V} \nabla \cdot (\rho \mathbf{v}) \, dV \\ &= \oint_{\partial V} \rho \mathbf{v} \cdot d\mathbf{S} \\ &= \int_{S_1} \rho \mathbf{v} \cdot d\mathbf{S} - \int_{S_2} \rho \mathbf{v} \cdot d\mathbf{S} = 0 \end{aligned} \\ \rho \mathbf{v} \cdot \mathbf{s} = \text{const.} \\ \begin{aligned}{} \rho \frac{D\mathbf{v}}{Dt} &= \rho \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} \\ &= \rho \nabla \left(\frac{v^2}{2}\right) - \rho \nabla \times \left( \nabla \times \mathbf{v} \right) \\ &= -\nabla \left( \rho gz \right) - \nabla p \end{aligned} \\ \nabla \left( p + \frac{1}{2} \rho v^2 + \rho gz \right) = \rho \mathbf{v} \times \left( \nabla \times \mathbf{v} \right) \\ \mathbf{v} \cdot \nabla \left( p + \frac{1}{2} \rho v^2 + \rho gz \right) = 0 \\ p + \frac{1}{2} \rho v^2 + \rho gz = \text{const.}\]

Special Relativity / 特殊相対性理論

Lorentz Transformation / Lorentz変換

\[c^2 dt^2 - dx^2 - dy^2 - dz^2 = \text{const.} \\ X' = LX \\ L = \begin{pmatrix} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & \frac{-\frac{v}{c}}{\sqrt{1 - \frac{v^2}{c^2}}} & 0 & 0 \\ \frac{-\frac{v}{c}}{\sqrt{1 - \frac{v^2}{c^2}}} & \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad X = \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix}\]

\[g = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad X = \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} \\ (dX)^T g (dX) = const. \\ (dX')^T g (dX') = (L dX)^T g (L dX) = (dX)^T g (dX) \\ L^T g L = g \\ L = \begin{pmatrix} \cosh \omega & \sinh \omega & 0 & 0 \\ \sinh \omega & \cosh \omega & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \\ \begin{pmatrix} ct' \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} \cosh \omega & \sinh \omega & 0 & 0 \\ \sinh \omega & \cosh \omega & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ vt \\ 0 \\ 0 \end{pmatrix} \\ \tanh \omega = -\frac{v}{c} \quad \cosh \omega = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad \sinh \omega = \frac{-\frac{v}{c}}{\sqrt{1 - \frac{v^2}{c^2}}}\]

Consequence of Transformation / 変換の帰結

\[\begin{aligned}{} \tau &= \frac{\tau_0}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \ell &= \ell_0 \sqrt{1 - \frac{v^2}{c^2}} \\ v_x &= \frac{v_x' + v}{1 + \frac{v}{c^2} v_x'} \\ v_y &= \frac{\sqrt{1 - \frac{v^2}{c^2}} v_y'}{1 + \frac{v}{c^2} v_x'} \end{aligned}\]

\[\Delta x' = 0 \quad \Delta t' = \tau_0 \\ \Delta t = \frac{\Delta t' - \frac{v}{c^2} \Delta x'}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \Delta t = 0 \quad \Delta x' = \ell_0 \\ \Delta x' = \frac{\Delta x - v \Delta t}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\Delta x}{\sqrt{1 - \frac{v^2}{c^2}}} \\ v_x = \frac{dx}{dt} = \frac{dx' + v dt'}{dt' + \frac{v}{c^2} dx'} = \frac{v_x' + v}{1 + \frac{v}{c^2} v_x'} \\ v_y = \frac{dy}{dt} = \frac{\sqrt{1 - \frac{v^2}{c^2}} \, dy'}{dt' + \frac{v}{c^2} dx'} = \frac{\sqrt{1 - \frac{v^2}{c^2}} v_y'}{1 + \frac{v}{c^2} v_x'}\]

Relativistic Dynamics / 相対論的力学

\[\delta S = \delta \int -mc \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2} = 0 \\ \begin{aligned}{} \mathbf{p} &= \frac{m\mathbf{v}}{\sqrt{1 - \frac{v^2}{c^2}}} \\ E &= \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} = \sqrt{p^2 c^2 + m^2 c^4} \end{aligned}\]

\[\begin{aligned}{} S &= \int -mc \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2} \\ &= \int -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \, dt \end{aligned} \\ L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \\ \mathbf{p} = \frac{\partial L}{\partial \mathbf{v}} = \frac{m\mathbf{v}}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \begin{aligned}{} E &= \mathbf{p} \cdot \mathbf{v} - L \\ &= \frac{m v^2}{\sqrt{1 - \frac{v^2}{c^2}}} + mc^2 \sqrt{1 - \frac{v^2}{c^2}} \\ &= \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} \end{aligned} \\ p^2 c^2 + m^2 c^4 = \frac{m^2 v^2 c^2}{1 - \frac{v^2}{c^2}} + \frac{m^2 c^4 - m^2 c^2 v^2}{1 - \frac{v^2}{c^2}} = E^2\]

Four-Vector / ４元ベクトル

\[\mathbf{A'} = L \mathbf{A} \\ \begin{aligned}{} \mathbf{X} &= (ct, x, y, z) \\ \mathbf{V} &= (\frac{c}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_x}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}}) \\ \mathbf{P} &= (\frac{E}{c}, p_x, p_y, p_z) \end{aligned}\]

\[\begin{aligned}{} d\mathbf{X} &= (cdt, dx, dy, dz) \\ d\tau &= \sqrt{dt^2 - \frac{dx^2 + dy^2 + dz^2}{c^2}} = dt \sqrt{1 - \frac{v^2}{c^2}} = \text{const.} \end{aligned} \\ \begin{aligned}{} \mathbf{V} &= \frac{d\mathbf{X}}{d\tau} = (\frac{c}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_x}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}}) \\ \mathbf{P} &= m\mathbf{V} = (\frac{E}{c}, p_x, p_y, p_z) \end{aligned}\]

Electromagnetism / 電磁気学

Maxwell’s Equation / Maxwell方程式

Charge and Current / 電荷と電流

\[\rho = \frac{dq}{d\tau} \quad \mathbf{J} = \rho \mathbf{v} \\ \frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{J}\]

\[\begin{aligned}{} \frac{d q}{d t} &= \frac{d}{d t} \int_V \rho \, d\tau = \int_V \left(\frac{\partial \rho}{\partial t}\right) d\tau \\ &= -\oint_S \mathbf{J} \cdot d\mathbf{a} = \int_V (-\nabla \cdot \mathbf{J}) \, d\tau \\ \end{aligned}\]

Electromagnetic Field / 電磁場

\[\begin{aligned}{} & \nabla \cdot \mathbf{E} = \frac{1}{\varepsilon_0} \rho \\ & \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ & \nabla \cdot \mathbf{B} = 0 \\ & \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned} \\ \mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}\]

Electromagnetic Potential / 電磁ポテンシャル

\[\begin{aligned}{} \mathbf{E} &= -\nabla V - \frac{\partial \mathbf{A}}{\partial t} \\ \mathbf{B} &= \nabla \times \mathbf{A} \end{aligned} \\ \begin{aligned}{} \nabla^2 V - \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} &= -\frac{1}{\varepsilon_0} \rho \\ \nabla^2 \mathbf{A} - \mu_0 \varepsilon_0 \frac{\partial \mathbf{A}}{\partial t} &= -\mu_0 \mathbf{J} \end{aligned} \\ \frac{d}{d t}(\mathbf{p} + q \mathbf{A}) = -\nabla (q V - q \mathbf{v} \cdot \mathbf{A})\]

\[\nabla \cdot \mathbf{B} = 0 \\ \mathbf{B} = \nabla \times \mathbf{A} \\ \nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{E} + \frac{\partial \mathbf{A}}{\partial t}\right) = 0 \\ \mathbf{E} + \frac{\partial \mathbf{A}}{\partial t} = -\nabla V \\ \nabla \cdot \mathbf{E} = -\nabla^2 V - \nabla \cdot \left(\frac{\partial \mathbf{A}}{\partial t}\right) = \frac{1}{\varepsilon_0} \rho \\ \nabla^2 V + \frac{\partial}{\partial t}(\nabla \cdot \mathbf{A}) = -\frac{1}{\varepsilon_0} \rho \\ \nabla \times \mathbf{B} - \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} = \nabla \times (\nabla \times \mathbf{A}) + \mu_0 \varepsilon_0 \frac{\partial}{\partial t} \left(\nabla V\right) + \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2} = \mu_0 \mathbf{J} \\ \left(\nabla^2 \mathbf{A} - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{A}}{\partial t^2}\right) - \nabla \left(\nabla \cdot \mathbf{A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t}\right) = -\mu_0 \mathbf{J} \\ \mathbf{A}' = \mathbf{A} + \nabla \lambda \quad V' = V - \frac{\partial \lambda}{\partial t} \\ \mathbf{B}' = \nabla \times \mathbf{A}' + \nabla \times \nabla \lambda = \nabla \times \mathbf{A} = \mathbf{B} \\ \mathbf{E}' = -\nabla V + \nabla \left(\frac{\partial \lambda}{\partial t}\right) - \frac{\partial \mathbf{A}}{\partial t} - \frac{\partial}{\partial t}(\nabla \lambda) = -\nabla V - \frac{\partial \mathbf{A}}{\partial t} = \mathbf{E} \\ \nabla^2 \lambda - \mu_0 \varepsilon_0 \frac{\partial^2 \lambda}{\partial t^2} = -\nabla \cdot \mathbf{A} - \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} \\ \nabla \cdot (\mathbf{A} + \nabla \lambda) = -\mu_0 \varepsilon_0 \frac{\partial}{\partial t}\left(V - \frac{\partial \lambda}{\partial t}\right) \\ \nabla \cdot \mathbf{A}' = -\mu_0 \varepsilon_0 \frac{\partial V'}{\partial t} \\ \begin{aligned}{} \nabla^2 V - \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} &= -\frac{1}{\varepsilon_0} \rho \\ \nabla^2 \mathbf{A} - \mu_0 \varepsilon_0 \frac{\partial \mathbf{A}}{\partial t} &= -\mu_0 \mathbf{J} \end{aligned} \\ \begin{aligned}{} \mathbf{F} = \frac{d \mathbf{p}}{d t} &= q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \\ &= q \left[-\nabla V - \frac{\partial \mathbf{A}}{\partial t} + \mathbf{v} \times (\nabla \times \mathbf{A})\right] \\ &= q \left[-\frac{\partial \mathbf{A}}{\partial t} - (\mathbf{v} \cdot \nabla)\mathbf{A} - \nabla V + \nabla(\mathbf{v} \cdot \mathbf{A})\right] \\ &= q \left[-\frac{d \mathbf{A}}{d t} - \nabla(V - \mathbf{v} \cdot \mathbf{A})\right] \end{aligned} \\ \frac{d}{d t}(\mathbf{p} + q \mathbf{A}) = -\nabla(q V - q \mathbf{v} \cdot \mathbf{A})\]

Energy Conservation / エネルギー保存

\[E_{\text{em}} = \int_V \left(\frac{\varepsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right) d\tau \\ \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \\ \frac{d}{dt}(E_{\text{mech}} + E_{\text{em}}) = -\oint_S \mathbf{S} \cdot d\mathbf{a}\]

\[\mathbf{F} \cdot d\mathbf{l} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \mathbf{v} \,dt = \mathbf{E} \cdot (q\mathbf{v}) \,dt \\ \frac{dE_{\text{mech}}}{dt} = \int_V \mathbf{E} \cdot \mathbf{J} \, d\tau \\ \begin{aligned}{} \mathbf{E} \cdot \mathbf{J} &= \frac{1}{\mu_0} \mathbf{E} \cdot (\nabla \times \mathbf{B}) - \varepsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \\ &= \frac{1}{\mu_0} [\mathbf{B} \cdot (\nabla \times \mathbf{E}) - \nabla \cdot (\mathbf{E} \times \mathbf{B})] - \varepsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} \\ &= -\varepsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} - \frac{1}{\mu_0} \mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} - \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B}) \\ &= -\frac{\partial}{\partial t}\left(\frac{\varepsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right) - \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B}) \\ \end{aligned} \\ \begin{aligned}{} \frac{dE_{\text{mech}}}{dt} &= - \int_V \frac{\partial}{\partial t} \left(\frac{\varepsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right) d\tau - \int_V \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B}) d\tau \\ &= -\frac{d}{d t} \int_V \left(\frac{\varepsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right) d\tau - \oint_S \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B}) \cdot d\mathbf{a} \\ &= -\frac{dE_{\text{em}}}{dt} - \oint_S \mathbf{S} \cdot d\mathbf{a} \end{aligned}\]

Momentum Conservation / 運動量保存

\[\mathbf{P}_{\text{em}} = \int_V \varepsilon_0 \mathbf{E} \times \mathbf{B} \, d\tau \\ T_{ij} = \varepsilon_0\left(E_i E_j - \frac{1}{2} \delta_{ij} E^2\right) + \frac{1}{\mu_0}\left(B_i B_j - \frac{1}{2} \delta_{ij} B^2\right) \\ \frac{d}{dt}(\mathbf{P}_{\text{mech}} + \mathbf{P}_{\text{em}}) = \oint_S \mathbf{T} \cdot d\mathbf{a}\]

\[\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B} \\ \frac{d\mathbf{P}_{\text{mech}}}{dt} = \int_V (\rho\mathbf{E} + \mathbf{J} \times \mathbf{B}) \, d\tau \\ \begin{aligned}{} & \rho\mathbf{E} + \mathbf{J} \times \mathbf{B} \\ =& \varepsilon_0(\nabla \cdot \mathbf{E})\mathbf{E} + \left(\frac{1}{\mu_0}\nabla \times \mathbf{B} - \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \times \mathbf{B} \\ =& \varepsilon_0(\nabla \cdot \mathbf{E})\mathbf{E} - \frac{1}{\mu_0} \mathbf{B} \times (\nabla \times \mathbf{B}) - \varepsilon_0 \mathbf{E} \times (\nabla \times \mathbf{E}) - \varepsilon_0 \frac{\partial}{\partial t} (\mathbf{E} \times \mathbf{B}) \\ =& \varepsilon_0(\nabla \cdot \mathbf{E})\mathbf{E} - \frac{1}{2\mu_0} \nabla(B^2) + \frac{1}{\mu_0} (\mathbf{B} \cdot \nabla)\mathbf{B} \\ &- \frac{\varepsilon_0}{2}\nabla(E^2) + \varepsilon_0(\mathbf{E} \cdot \nabla)\mathbf{E} - \varepsilon_0 \frac{\partial}{\partial t} (\mathbf{E} \times \mathbf{B}) \\ =& \varepsilon_0\left[(\nabla \cdot \mathbf{E})\mathbf{E} + (\mathbf{E} \cdot \nabla)\mathbf{E}\right] + \frac{1}{\mu_0}\left[(\nabla \cdot \mathbf{B})\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{B}\right] \\ &- \nabla\left(\frac{\varepsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right) - \varepsilon_0 \frac{\partial}{\partial t} (\mathbf{E} \times \mathbf{B}) \\ =& \nabla \cdot \overleftrightarrow{T} - \frac{\partial}{\partial t} (\varepsilon_0 \mathbf{E} \times \mathbf{B}) \\ \end{aligned} \\ \begin{aligned}{} \frac{d\mathbf{P}_{\text{mech}}}{dt} &= - \int_V \frac{\partial}{\partial t} (\varepsilon_0 \mathbf{E} \times \mathbf{B}) d\tau + \int_V \nabla \cdot \overleftrightarrow{T} d\tau \\ &= -\frac{d}{d t} \int_V \varepsilon_0 \mathbf{E} \times \mathbf{B} d\tau + \oint_S \overleftrightarrow{T} \cdot d\mathbf{a} \\ &= -\frac{d\mathbf{P}_{\text{em}}}{dt} + \oint_S \overleftrightarrow{T} \cdot d\mathbf{a} \end{aligned}\]

Electrostatics / 静電場

Electric Field / 電場

\[\begin{aligned}{} & \nabla \cdot \mathbf{E} = \frac{1}{\varepsilon_0} \rho && \oint_S \mathbf{E} \cdot d\mathbf{a} = \frac{1}{\varepsilon_0} Q \\ & \nabla \times \mathbf{E} = 0 && \oint_C \mathbf{E} \cdot d\mathbf{l} = 0 \\ \end{aligned} \\ \mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\mathbf{r}')\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \, d\tau'\]

\[\begin{aligned}{} \nabla \cdot \mathbf{E}(\mathbf{r}) &= \frac{1}{4\pi\varepsilon_0} \int_V \nabla \cdot \left(\frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2}\right) \rho(\mathbf{r}') \, d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \int_V 4\pi\delta^3(\mathbf{r}-\mathbf{r}') \rho(\mathbf{r}') \, d\tau' \\ &= \frac{1}{\varepsilon_0} \rho(\mathbf{r}) \end{aligned} \\ \begin{aligned}{} \nabla \times \mathbf{E}(\mathbf{r}) &= \frac{1}{4\pi\varepsilon_0} \int_V \nabla \times \left(\frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2}\right) \rho(\mathbf{r}') \, d\tau' \\ &= 0 \end{aligned}\]

Electric Potential / 電位

\[\mathbf{E} = -\nabla V \\ \nabla^2 V = -\frac{1}{\varepsilon_0} \rho \\ V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\mathbf{r}')}{\vec{r}} d\tau'\]

\[\nabla \times \mathbf{E} = -\nabla \times \nabla V = 0 \\ \nabla \cdot \mathbf{E} = -\nabla^2 V = \frac{1}{\varepsilon_0} \rho \\ \begin{aligned}{} \nabla^2 V(\mathbf{r}) &= \frac{1}{4\pi\varepsilon_0} \int_V \nabla^2 \left(\frac{1}{\vec{r}}\right) \rho(\mathbf{r}') d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \int_V -4\pi \delta^3(\mathbf{r}-\mathbf{r}') \rho(\mathbf{r}') d\tau' \\ &= -\frac{1}{\varepsilon_0} \rho(\mathbf{r}) \end{aligned}\]

Electric Energy / 電気エネルギー

\[\begin{aligned}{} \Delta W &= q \Delta V \\ W &= \frac{1}{2} \int_V \rho V \, d\tau \\ W &= \frac{\varepsilon_0}{2} \int_V E^2 \, d\tau \end{aligned}\]

\[\Delta W = \int_a^b \mathbf{F} \cdot d\mathbf{l} = \int_a^b q\mathbf{E} \cdot d\mathbf{l} = q \Delta V \\ \begin{aligned}{} W &= \sum_i \sum_{j<i} q_i V_{ij} \\ &= \frac{1}{2} \sum_i \sum_{j \neq i} q_i V_{ij} \\ &= \frac{1}{2} \sum_i q_i V_i \\ &= \frac{1}{2} \int_V \rho V \, d\tau \\ &= \frac{\varepsilon_0}{2} \int_V (\nabla \cdot \mathbf{E}) V \, d\tau \\ &= \frac{\varepsilon_0}{2} \left[ \oint_S (\mathbf{E}V) \cdot d\mathbf{a} - \int_V \mathbf{E} \cdot (\nabla V) \, d\tau \right] \\ &= \frac{\varepsilon_0}{2} \int_V E^2 \, d\tau \end{aligned}\]

Electric Dipole / 電気双極子

\[V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int_V (r')^n P_n(\cos\alpha) \rho(\mathbf{r}') d\tau' \\ p = qd \\ V_{\text{dip}} = \frac{1}{4\pi\varepsilon_0} \frac{p\cos\theta}{r^2} \\ \mathbf{E}_{\text{dip}} = \frac{1}{4\pi\varepsilon_0} \frac{p(2\cos\theta \,\hat{\mathbf{r}} + \sin\theta \,\hat{\boldsymbol{\theta}})}{r^3}\]

\[\frac{1}{\vec{r}} = \frac{1}{\sqrt{r^2 + (r')^2 - 2rr'\cos\alpha}} = \frac{1}{r} \sum_{n=0}^{\infty} \left(\frac{r'}{r}\right)^n P_n(\cos\alpha) \\ \begin{aligned}{} V(\mathbf{r}) &= \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(\mathbf{r}')}{\vec{r}} d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \int_V \sum_{n=0}^{\infty} \frac{(r')^n}{r^{n+1}} P_n(\cos\alpha) \rho(\mathbf{r}') d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int_V (r')^n P_n(\cos\alpha) \rho(\mathbf{r}') d\tau' \end{aligned} \\ \begin{aligned}{} V_{\text{dip}} &= \frac{1}{4\pi\varepsilon_0} \frac{1}{r^2} \int_V r' (\cos\alpha) \rho(\mathbf{r}') d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \frac{1}{r^2} \int_V (\mathbf{r}' \cdot \hat{\mathbf{r}}) \rho(\mathbf{r}') d\tau' \\ &= \frac{1}{4\pi\varepsilon_0} \frac{q \mathbf{d} \cdot \hat{\mathbf{r}}}{r^2} \\ &= \frac{1}{4\pi\varepsilon_0} \frac{qd\cos\theta}{r^2} \end{aligned} \\ \begin{aligned}{} \mathbf{E}_{\text{dip}} &= -\nabla V_{\text{dip}} \\ &= \frac{1}{4\pi\varepsilon_0} \frac{qd(2\cos\theta \,\hat{\mathbf{r}} + \sin\theta \,\hat{\boldsymbol{\theta}})}{r^3} \end{aligned}\]

Uniqueness Theorem / 一意性定理

\[\begin{aligned}{} &\rho_1^{\text{volume}} = \rho_2^{\text{volume}}, && V_1^{\text{boundary}} = V_2^{\text{boundary}} \rightarrow &&V_1^{\text{volume}} = V_2^{\text{volume}} \\ & \rho_1^{\text{volume}} = \rho_2^{\text{volume}}, && Q_1^{\text{boundary}} = Q_2^{\text{boundary}} \rightarrow &&\mathbf{E}_1^{\text{volume}} = \mathbf{E}_2^{\text{volume}} \end{aligned}\]

\[\nabla^2 V_3 = \nabla^2 V_1 - \nabla^2 V_2 = -\frac{\rho_1}{\varepsilon_0} + \frac{\rho_2}{\varepsilon_0} = 0 \\ V_3^\Sigma = V_1^\Sigma - V_2^\Sigma = 0 \\ V_3 = 0 \rightarrow V_1 = V_2 \\ \nabla \cdot \mathbf{E}_3 = \nabla \cdot \mathbf{E}_1 - \nabla \cdot \mathbf{E}_2 = \frac{\rho_1}{\varepsilon_0} - \frac{\rho_2}{\varepsilon_0} = 0 \\ \oint_S \mathbf{E}_3 \cdot d\mathbf{a} = \oint_S \mathbf{E}_1 \cdot d\mathbf{a} - \oint_S \mathbf{E}_2 \cdot d\mathbf{a} = \frac{Q_1}{\varepsilon_0} - \frac{Q_2}{\varepsilon_0} = 0 \\ \nabla \cdot (V_3 \mathbf{E}_3) = (\nabla V_3) \cdot \mathbf{E}_3 + V_3 (\nabla \cdot \mathbf{E}_3) = -{E_3}^2 \\ \begin{aligned}{} \int_V \nabla \cdot (V_3 \mathbf{E}_3) d\tau &= \oint_S (V_3 \mathbf{E}_3) \cdot d\mathbf{a} = \sum V_3 \oint_S \mathbf{E}_3 \cdot d\mathbf{a} \\ &= 0 = -\int_V {E_3}^2 d\tau \end{aligned}\\ \mathbf{E}_3 = 0 \rightarrow \mathbf{E}_1 = \mathbf{E}_2\]

Capacitor / コンデンサー

\[\begin{aligned}{} Q &= CV \\ I &= C \frac{dV}{dt} \\ W &= \frac{1}{2}CV^2 \\ \end{aligned}\]

\[I = \frac{dQ}{dt} = C \frac{dV}{dt} \\ dW = dQ \cdot V = CV \, dV \\ W = \frac{1}{2}CV^2\]

Magnetostatics / 静磁場

Magnetic Field / 磁場

\[\begin{aligned}{} & \nabla \cdot \mathbf{B} = 0 && \oint_S \mathbf{B} \cdot d\mathbf{a} = 0 \\ & \nabla \times \mathbf{B} = \mu_0 \mathbf{J} && \oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \end{aligned} \\ \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{J}(\mathbf{r}') \times \hat{\mathbf{\vec{r}}}}{\vec{r}^2} \, d\tau'\]

\[\begin{aligned}{} \nabla \cdot \mathbf{B}(\mathbf{r}) &= \frac{\mu_0}{4\pi} \int_V \nabla \cdot \left[ \mathbf{J}(\mathbf{r}') \times \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \right] \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V -\mathbf{J}(\mathbf{r}') \cdot \left( \nabla \times \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \right) \, d\tau' \\ &= 0 \end{aligned} \\ \begin{aligned}{} \nabla \times \mathbf{B}(\mathbf{r}) &= \frac{\mu_0}{4\pi} \int_V \nabla \times \left[ \mathbf{J}(\mathbf{r}') \times \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \right] \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \left[ \mathbf{J}(\mathbf{r}') (\nabla \cdot \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2}) - (\mathbf{J}(\mathbf{r}') \cdot \nabla) \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \right] \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \mathbf{J}(\mathbf{r}') 4\pi \delta^3(\mathbf{r} - \mathbf{r}') \, d\tau' + \frac{\mu_0}{4\pi} \int_V (\mathbf{J}(\mathbf{r}') \cdot \nabla') \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \, d\tau' \\ &= \mu_0 \mathbf{J}(\mathbf{r}) + \frac{\mu_0}{4\pi} \int_V \left[ \nabla'_2 \cdot \left( \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \otimes \mathbf{J}(\mathbf{r}') \right) - \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \left( \nabla' \cdot \mathbf{J}(\mathbf{r}') \right) \right] \, d\tau' \\ &= \mu_0 \mathbf{J}(\mathbf{r}) + \frac{\mu_0}{4\pi} \oint_S \left(\frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \otimes \mathbf{J}(\mathbf{r}')\right) \cdot d\mathbf{a}' + \frac{\mu_0}{4\pi} \int_V \frac{\hat{\mathbf{\vec{r}}}}{\vec{r}^2} \frac{\partial \rho(\mathbf{r}')}{\partial t} \, d\tau' \\ &= \mu_0 \mathbf{J}(\mathbf{r}) \end{aligned}\]

Magnetic Potential / 磁位

\[\mathbf{B} = \nabla \times \mathbf{A} \\ \nabla^2 \mathbf{A} = -\mu_0 \mathbf{J} \\ \mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{J}(\mathbf{r}')}{\vec{r}} \, d\tau'\]

\[\nabla \cdot \mathbf{B} = \nabla \cdot (\nabla \times \mathbf{A}) = 0 \\ \nabla \times \mathbf{B} = \nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \\ \mathbf{A} = \mathbf{A}_0 + \nabla \lambda \\ \mathbf{B} = \nabla \times \mathbf{A}_0 + \nabla \times \nabla \lambda = \nabla \times \mathbf{A}_0 = \mathbf{B}_0 \\ \nabla^2 \lambda = -\nabla \cdot \mathbf{A}_0 \\ \nabla \cdot \mathbf{A} = \nabla \cdot \mathbf{A}_0 + \nabla^2 \lambda = 0 \\ \nabla \times \mathbf{B} = -\nabla^2 \mathbf{A} = \mu_0 \mathbf{J} \\ \begin{aligned}{} \nabla^2 \mathbf{A}(\mathbf{r}) &= \frac{\mu_0}{4\pi} \int_V \nabla^2 \left( \frac{1}{\vec{r}} \right) \mathbf{J}(\mathbf{r}') \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V -4\pi \delta^3(\mathbf{r} - \mathbf{r}') \mathbf{J}(\mathbf{r}') \, d\tau' \\ &= -\mu_0 \mathbf{J}(\mathbf{r}) \end{aligned}\]

Magnetic Energy / 磁気エネルギー

\[W = \frac{1}{2} \int_V \mathbf{J} \cdot \mathbf{A} \, d\tau \\ W = \frac{1}{2\mu_0} \int_V B^2 \, d\tau\]

\[\begin{aligned}{} \delta W &= - \delta q \oint_C \mathbf{E} \cdot d\mathbf{l} \\ &= - \delta q \oint_S (\nabla \times \mathbf{E}) \cdot d\mathbf{a} \\ &= I \delta t \oint_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{a} \\ &= I \oint_S \delta \mathbf{B} \cdot d\mathbf{a} \\ &= I \oint_S (\nabla \times \delta \mathbf{A}) \cdot d\mathbf{a} \\ &= I \oint_C \delta \mathbf{A} \cdot d\mathbf{l} \\ &= \int_C \mathbf{I} \cdot \delta \mathbf{A} \, dl \\ &= \int_V \mathbf{J} \cdot \delta \mathbf{A} \, d\tau \\ &= \int_V \frac{1}{2} \delta (\mathbf{J} \cdot \mathbf{A}) \, d\tau \end{aligned} \\ \begin{aligned}{} W &= \frac{1}{2} \int_V \mathbf{J} \cdot \mathbf{A} \, d\tau \\ &= \frac{1}{2\mu_0} \int_V (\nabla \times \mathbf{B}) \cdot \mathbf{A} \, d\tau \\ &= \frac{1}{2\mu_0} \left[ \int_V \mathbf{B} \cdot (\nabla \times \mathbf{A}) \, d\tau - \int_V \nabla \cdot (\mathbf{A} \times \mathbf{B}) \, d\tau \right] \\ &= \frac{1}{2\mu_0} \int_V B^2 \, d\tau - \frac{1}{2\mu_0} \oint_S (\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{a} \\ &= \frac{1}{2\mu_0} \int_V B^2 \, d\tau \end{aligned}\]

Magnetic Dipole / 磁気双極子

\[\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int_V (r')^n P_n(\cos\alpha) \mathbf{J}(\mathbf{r}') \, d\tau' \\ m = I a \\ \mathbf{A}_{\text{dip}} = \frac{\mu_0}{4\pi} \frac{m \sin\theta \, \hat{\boldsymbol{\phi}}}{r^2} \\ \mathbf{B}_{\text{dip}} = \frac{\mu_0}{4\pi} \frac{m( 2\cos\theta \, \hat{\mathbf{r}} + \sin\theta \, \hat{\boldsymbol{\theta}} )}{r^3}\]

\[\frac{1}{\vec{r}} = \frac{1}{\sqrt{r^2 + (r')^2 - 2rr'\cos\alpha}} = \frac{1}{r} \sum_{n=0}^{\infty} \left(\frac{r'}{r}\right)^n P_n(\cos\alpha) \\ \begin{aligned}{} \mathbf{A}(\mathbf{r}) &= \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{J}(\mathbf{r}')}{\vec{r}} \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \sum_{n=0}^{\infty} \frac{(r')^n}{r^{n+1}} P_n(\cos\alpha) \mathbf{J}(\mathbf{r}') \, d\tau' \\ &= \frac{\mu_0}{4\pi} \sum_{n=0}^{\infty} \frac{1}{r^{n+1}} \int_V (r')^n P_n(\cos\alpha) \mathbf{J}(\mathbf{r}') \, d\tau' \end{aligned} \\ \begin{aligned}{} \mathbf{A}_{\text{dip}} &= \frac{\mu_0}{4\pi} \frac{1}{r^2} \oint_C r' (\cos\alpha) I \,d\mathbf{l}' \\ &= \frac{\mu_0}{4\pi} \frac{I}{r^2} \oint_C \mathbf{r}' \cdot \hat{\mathbf{r}} \,d\mathbf{l}' \\ &= \frac{\mu_0}{4\pi} \frac{I\mathbf{a} \times \hat{\mathbf{r}}}{r^2} \\ &= \frac{\mu_0}{4\pi} \frac{I a \sin\theta \, \hat{\boldsymbol{\phi}}}{r^2} \end{aligned} \\ \begin{aligned}{} \mathbf{B}_{\text{dip}} &= \nabla \times \mathbf{A}_{\text{dip}} \\ &= \frac{\mu_0}{4\pi} \frac{I a ( 2\cos\theta \, \hat{\mathbf{r}} + \sin\theta \, \hat{\boldsymbol{\theta}} )}{r^3} \end{aligned}\]

Electromagnetic Induction / 電磁誘導

\[\begin{aligned}{} & \mathcal{E} = \int_C (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l} \\ & \mathcal{E} = - \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{a} \end{aligned}\]

\[\begin{aligned}{} \mathcal{E} &= \frac{1}{q} \int_C q \, (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l} \\ &= \int_C (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l} \\ \mathcal{E} &= \frac{1}{q} \oint_C q\mathbf{E} \cdot d\mathbf{l} \\ &= \int_S (\nabla \times \mathbf{E}) \cdot d\mathbf{a} \\ &= \int_S -\frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{a} \\ &= - \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{a} \end{aligned}\]

Inductor / インダクタ

\[\begin{aligned}{} & \Phi = L I \\ & V = L \frac{dI}{dt} \\ & W = \frac{1}{2} L I^2 \end{aligned}\]

\[\begin{aligned}{} V &= \frac{d\Phi}{dt} = L \frac{dI}{dt} \\ dW &=I\,dt \cdot L \frac{dI}{dt} = L I \, dI \\ W &= \frac{1}{2} L I^2 \end{aligned}\]

Electromagnetism in Matter / 物質中の電磁気学

Polarization / 分極

\[\begin{aligned}{} \mathbf{N} &= \mathbf{p} \times \mathbf{E} \\ \mathbf{F} &= \nabla (\mathbf{p} \cdot \mathbf{E}) \end{aligned}\]

\[\begin{aligned}{} \mathbf{N} &= \int_V \mathbf{r}' \times \left( \rho(\mathbf{r}') \, d\tau' \mathbf{E} \right) \\ &= \left( \int_V \mathbf{r}' \rho(\mathbf{r}') \, d\tau' \right) \times \mathbf{E} \end{aligned} \\ \begin{aligned}{} \mathbf{F} &= \int_V \rho(\mathbf{r}') \, d\tau' \, \mathbf{E} \\ &= \int_V \rho(\mathbf{r}') \, d\tau' \left[ \mathbf{E}_0 + \mathbf{r}' \cdot (\nabla \mathbf{E})_0 \right] \\ &= \left( \int_V \rho(\mathbf{r}') \, d\tau' \right) \mathbf{E}_0 + \left( \int_V \mathbf{r}' \rho(\mathbf{r}') \, d\tau' \right) \cdot \nabla \mathbf{E} \\ &= \nabla \left[ \left( \int_V \mathbf{r}' \rho(\mathbf{r}') \, d\tau' \right) \cdot \mathbf{E} \right] \end{aligned}\]

Electric Displacement / 電束密度

\[\rho_b = -\nabla \cdot \mathbf{P} \quad \sigma_b = \mathbf{P} \cdot \hat{\mathbf{n}} \\ \mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon \mathbf{E} \\ \nabla \cdot \mathbf{D} = \rho_f\]

\[\begin{aligned}{} V &= \frac{1}{4\pi \varepsilon_0} \int_V \frac{\mathbf{P}(\mathbf{r}') \cdot \hat{\mathbf{\vec{r}}}}{\vec{r}^2} \, d\tau' \\ &= \frac{1}{4\pi \varepsilon_0} \int_V \mathbf{P}(\mathbf{r}') \cdot \nabla' \left(\frac{1}{\vec{r}}\right) \, d\tau' \\ &= \frac{1}{4\pi \varepsilon_0} \int_V \left[-\frac{1}{\vec{r}}\nabla' \cdot \mathbf{P}(\mathbf{r}') + \nabla' \cdot \left(\frac{\mathbf{P}(\mathbf{r}')}{\vec{r}}\right)\right] d\tau' \\ &= \frac{1}{4\pi \varepsilon_0} \int_V \frac{-\nabla' \cdot \mathbf{P}(\mathbf{r}')}{\vec{r}} \, d\tau' + \frac{1}{4\pi \varepsilon_0} \oint_S \frac{\mathbf{P}(\mathbf{r}')}{\vec{r}} \cdot d\mathbf{a}' \\ &= \frac{1}{4\pi \varepsilon_0} \int_V \frac{\rho_b(\mathbf{r}')}{\vec{r}} \, d\tau' + \frac{1}{4\pi \varepsilon_0} \oint_S \frac{\sigma_b(\mathbf{r}')}{\vec{r}} \, da' \end{aligned} \\ \nabla \cdot \mathbf{E} = \frac{1}{\varepsilon_0} \left(\rho_f - \nabla \cdot \mathbf{P}\right) \\ \nabla \cdot \left(\varepsilon_0 \mathbf{E} + \mathbf{P}\right) = \rho_f\]

Magnetization / 磁化

\[\begin{aligned}{} \mathbf{N} &= \mathbf{m} \times \mathbf{B} \\ \mathbf{F} &= \nabla (\mathbf{m} \cdot \mathbf{B}) \end{aligned}\]

\[\begin{aligned}{} \mathbf{N} &= \oint_C \mathbf{r}' \times \left( I \, d\mathbf{l}' \times \mathbf{B} \right) \\ &= I \oint_C d\mathbf{l}' \left( \mathbf{r}' \cdot \mathbf{B} \right) - I \oint_C \mathbf{B} \left( \mathbf{r}' \cdot d\mathbf{l}' \right) \\ &= (I \, \mathbf{a}) \times \mathbf{B} \end{aligned} \\ \begin{aligned}{} \mathbf{F} &= \oint_C I \, d\mathbf{l}' \times \mathbf{B} \\ &= I \oint_C d\mathbf{l}' \times \left[ \mathbf{B}_0 + \mathbf{r}' \cdot (\nabla \mathbf{B})_0 \right] \\ &= \left( I \oint_C d\mathbf{l}' \right) \times \mathbf{B}_0 + I \oint_C d\mathbf{l}' \times \left[ \mathbf{r}' \cdot (\nabla \mathbf{B})_0 \right] \\ &= \nabla \left[ \left(I \, \mathbf{a} \right) \cdot \mathbf{B} \right] \end{aligned}\]

Auxiliary Field / 補助磁場

\[\mathbf{J}_b = \nabla \times \mathbf{M} \quad \mathbf{K}_b = \mathbf{M} \times \hat{\mathbf{n}} \\ \mathbf{H} = \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} = \frac{1}{\mu} \mathbf{B} \\ \nabla \times \mathbf{H} = \mathbf{J}_f\]

\[\begin{aligned}{} \mathbf{A}(\mathbf{r}) &= \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{M}(\mathbf{r}') \times \hat{\mathbf{\vec{r}}}}{\vec{r}^2} \, d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \mathbf{M}(\mathbf{r}') \times \nabla' \left(\frac{1}{\vec{r}}\right) d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \left[ \frac{1}{\vec{r}}\nabla' \times \mathbf{M}(\mathbf{r}') - \nabla' \times \left(\frac{\mathbf{M}(\mathbf{r}')}{\vec{r}}\right) \right] d\tau' \\ &= \frac{\mu_0}{4\pi} \int_V \frac{\nabla' \times \mathbf{M}(\mathbf{r}')}{\vec{r}} \, d\tau' + \frac{\mu_0}{4\pi} \oint_S \frac{\mathbf{M}(\mathbf{r}')}{\vec{r}} \times d\mathbf{a}' \\ &= \frac{\mu_0}{4\pi} \int_V \frac{\mathbf{J}_b(\mathbf{r}')}{\vec{r}} \, d\tau' + \frac{\mu_0}{4\pi} \oint_S \frac{\mathbf{K}_b(\mathbf{r}')}{\vec{r}} \, da \end{aligned} \\ \nabla \times \mathbf{B} = \mu_0 (\mathbf{J}_f + \nabla \times \mathbf{M}) \\ \nabla \times \left(\frac{1}{\mu_0} \mathbf{B} - \mathbf{M}\right) = \mathbf{J}_f\]

Maxwell’s Equations in Matter / 物質中のMaxwell方程式

\[\begin{aligned}{} & \nabla \cdot \mathbf{D} = \rho_f \\ & \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ & \nabla \cdot \mathbf{B} = 0 \\ & \nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t} \end{aligned}\]

\[\begin{aligned}{} \rho &= \rho_f + \rho_b \\ &= \rho_f - \nabla \cdot \mathbf{P} \end{aligned} \\ \begin{aligned}{} \mathbf{J} &= \mathbf{J}_f + \mathbf{J}_b + \mathbf{J}_p \\ &= \mathbf{J}_f + \nabla \times \mathbf{M} + \frac{\partial \mathbf{P}}{\partial t} \end{aligned} \\ \nabla \cdot \mathbf{E} = \frac{1}{\varepsilon_0} \left( \rho_f - \nabla \cdot \mathbf{P} \right) \\ \nabla \cdot \left( \varepsilon_0 \mathbf{E} + \mathbf{P} \right) = \rho_f \\ \nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J}_f + \nabla \times \mathbf{M} + \frac{\partial \mathbf{P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \nabla \times \left( \frac{1}{\mu_0} \mathbf{B} - \mathbf{M} \right) = \mathbf{J}_f + \frac{\partial}{\partial t} \left( \varepsilon_0 \mathbf{E} + \mathbf{P} \right)\]

Boundary Conditions / 境界条件

\[\begin{aligned}{} & \mathbf{D}_{1n} - \mathbf{D}_{2n} = \sigma_f \,\hat{\mathbf{n}} \\ & \mathbf{E}_{1t} - \mathbf{E}_{2t} = 0 \\ & \mathbf{B}_{1n} - \mathbf{B}_{2n} = 0 \\ & \mathbf{H}_{1t} - \mathbf{H}_{2t} = \mathbf{K}_f \times \hat{\mathbf{n}} \end{aligned}\]

\[\begin{aligned}{} \oint_S \mathbf{D} \cdot d\mathbf{a} &= \mathbf{D}_{1n}\cdot \mathbf{a} - \mathbf{D}_{2n}\cdot \mathbf{a} \\ &= \sigma_f \,\hat{\mathbf{n}}\cdot \mathbf{a} \end{aligned} \\ \begin{aligned}{} \oint_C \mathbf{E} \cdot d\mathbf{l} &= \mathbf{E}_{1t}\cdot \mathbf{l} - \mathbf{E}_{2t}\cdot \mathbf{l} \\ &= 0 \end{aligned} \\ \begin{aligned}{} \oint_S \mathbf{B} \cdot d\mathbf{a} &= \mathbf{B}_{1n}\cdot \mathbf{a} - \mathbf{B}_{2n}\cdot \mathbf{a} \\ &= 0 \end{aligned} \\ \begin{aligned}{} \oint_C \mathbf{H} \cdot d\mathbf{l} &= \mathbf{H}_{1t}\cdot \mathbf{l} - \mathbf{H}_{2t}\cdot \mathbf{l} \\ &= \mathbf{K}_f \cdot (\hat{\mathbf{n}}\times \mathbf{l}) \\ &= (\mathbf{K}_f \times \hat{\mathbf{n}}) \cdot \mathbf{l} \end{aligned}\]

Electrodynamics / 電気力学

Electromagnetic Wave / 電磁波

Radiation / 放射

Electrical Circuit / 電気回路

Relativistic Electrodynamics / 相対論的電気力学

Relativistic Current / 相対論的電流

\[J^\mu = \left( c\rho, J_x, J_y, J_z \right) \\ \bar{J}^{\mu} = \Lambda^{\mu}_{\nu} J^{\nu} \\ \frac{\partial J^{\mu}}{\partial x^{\mu}} = 0\]

\[c\rho = \rho_0 \frac{c}{\sqrt{1 - \frac{u^2}{c^2}}} \\ \mathbf{J} = \rho \mathbf{v} = \rho_0 \frac{\mathbf{v}}{\sqrt{1 - \frac{v^2}{c^2}}} \\ J^{\mu} = \rho_0 v^{\mu} \\ \frac{\partial J^{\mu}}{\partial x^{\mu}} = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0\]

Relativistic Field / 相対論的場

\[F^{\mu \nu} = \begin{pmatrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ -\frac{E_x}{c} & 0 & B_z & -B_y \\ -\frac{E_y}{c} & -B_z & 0 & B_x \\ -\frac{E_z}{c} & B_y & -B_x & 0 \end{pmatrix} \\ G^{\mu \nu} = \begin{pmatrix} 0 & B_x & B_y & B_z \\ -B_x & 0 & -\frac{E_z}{c} & \frac{E_y}{c} \\ -B_y & \frac{E_z}{c} & 0 & -\frac{E_x}{c} \\ -B_z & -\frac{E_y}{c} & \frac{E_x}{c} & 0 \end{pmatrix} \\ \bar{F}^{\mu \nu} = \Lambda^\mu_{\lambda} \Lambda^\nu_{\sigma} F^{\lambda \sigma} \quad \bar{G}^{\mu \nu} = \Lambda^\mu_{\lambda} \Lambda^\nu_{\sigma} G^{\lambda \sigma} \\ \frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu \quad \frac{\partial G^{\mu \nu}}{\partial x^\nu} = 0 \\ \left( \frac{F}{\sqrt{1 - \frac{v^2}{c^2}}} \right)^\mu = q \left( \frac{v}{\sqrt{1 - \frac{v^2}{c^2}}} \right)_\nu F^{\mu \nu}\]

\[\begin{aligned}{} \frac{\partial F^{0\nu}}{\partial x^\nu} &= \frac{1}{c} \frac{\partial E_x}{\partial x} + \frac{1}{c} \frac{\partial E_y}{\partial y} + \frac{1}{c} \frac{\partial E_z}{\partial z} = \frac{1}{c} (\nabla \cdot \mathbf{E}) = \mu_0 c \rho \\ \frac{\partial F^{1\nu}}{\partial x^\nu} &= -\frac{1}{c^2} \frac{\partial E_x}{\partial t} + \frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z} = \left( -\frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} + \nabla \times \mathbf{B} \right)_x = \mu_0 J_x \\ \frac{\partial G^{0\nu}}{\partial x^\nu} &= \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z} = \nabla \cdot \mathbf{B} = 0 \\ \frac{\partial G^{1\nu}}{\partial x^\nu} &= -\frac{1}{c} \frac{\partial B_x}{\partial t} - \frac{1}{c} \frac{\partial E_z}{\partial y} + \frac{1}{c} \frac{\partial E_y}{\partial z} = -\frac{1}{c} \left( \frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathbf{E} \right)_x = 0 \\ \end{aligned} \\ \begin{aligned}{} \frac{F_x}{\sqrt{1 - \frac{v^2}{c^2}}} &= \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \left( q \mathbf{E} + q \mathbf{v} \times \mathbf{B} \right)_x \\ &= q \left[ \frac{-c}{\sqrt{1 - \frac{v^2}{c^2}}} \left( \frac{-E_x}{c} \right) + \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}} B_z - \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}} B_y \right] \end{aligned}\]

Relativistic Potential / 相対論的ポテンシャル

\[A^\mu = \left( \frac{V}{c}, A_x, A_y, A_z \right) \\ \bar{A}^\mu = \Lambda^\mu_{\nu} A^\nu \\ F^{\mu \nu} = \frac{\partial A^\nu}{\partial x_\mu} - \frac{\partial A^\mu}{\partial x_\nu} \\ \frac{\partial}{\partial x_\nu} \frac{\partial}{\partial x^\nu} A^\mu = -\mu_0 J^\mu\]

\[\begin{aligned}{} F^{01} &= \frac{E_x}{c} = -\frac{1}{c} \left( \frac{\partial \mathbf{A}}{\partial t} + \nabla V \right)_x = -\frac{1}{c} \frac{\partial A_x}{\partial t} - \frac{1}{c} \frac{\partial V}{\partial x} \\ F^{12} &= B_z = (\nabla \times \mathbf{A})_z = \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \end{aligned} \\ \begin{aligned}{} \left( -\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \nabla^2 \right) \frac{V}{c} &= -\mu_0 c \rho \\ \left( -\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \nabla^2 \right) \mathbf{A} &= -\mu_0 \mathbf{J} \\ \end{aligned}\]

Thermodynamics / 熱力学

First Law of Thermodynamics / 熱力学第一法則

Second Law of Thermodynamics / 熱力学第二法則

Thermodynamic Potential / 熱力学ポテンシャル

Microstate and Probability / 微視状態と確率

Ensemble Theory / アンサンブル理論

Classicial Ideal Gas / 古典理想気体

Quantum Ideal Gas / 量子理想気体

Lattice Vibration / 格子振動

Phase Transition / 相転移

Quantum Mechanics / 量子力学

\[\begin{aligned}{} & |\psi\rangle \in \mathcal{H} \\ & \hat{A} = \hat{A}^\dagger \\ & P(a_n) = |\langle a_n | \psi \rangle|^2 \\ & \hat{H}|\psi(t)\rangle = i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle \end{aligned}\]

Formalism / 形式論

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 / 物理量

\[\begin{aligned}{} & \hat{A}|a\rangle = a|a\rangle && \langle a | a \rangle = \mathbb I \\ & a = a^* && \langle a_n | a_m \rangle = \delta_{nm} \\ & \hat{A} = \sum_a a |a\rangle\langle a| && \hat{A} = \int a |a\rangle\langle a| \, da \end{aligned}\]

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

\[\begin{aligned}{} & |\psi(t)\rangle = \hat U(t)\,|\psi(0)\rangle \quad \hat U^\dagger(t)\,\hat U(t) = \mathbb I \\ & \hat U(t) = \exp\left[-\frac{i}{\hbar}\hat H t\right] \\ & \hat U(t) = \mathcal T \exp\left[-\frac{i}{\hbar}\int_{0}^{t}\hat H(t')\,dt'\right] \end{aligned}\]

Wave Mechanics / 波動力学

Wave Function / 波動関数

\[\begin{aligned}{} & \langle x | \psi \rangle = \psi(x) \\ & \langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \phi^*(x) \psi(x) dx \\ & \langle x | \hat{x} | \psi \rangle = x \psi(x) \\ & \langle x | \hat{p} | \psi \rangle = -i\hbar \frac{\partial}{\partial x} \psi(x) \end{aligned}\]

Change of Basis / 基底変換

\[\begin{aligned}{} & \langle x | x' \rangle = \delta(x - x') && \langle p | p' \rangle = \delta(p - p') \\ & \langle x | p \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{\frac{i}{\hbar}px} && \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-\frac{i}{\hbar}px} \\ & \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-\frac{i}{\hbar}px} dx && \psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{\frac{i}{\hbar}px} dp \end{aligned}\]

Schrödinger Equation / Schrödinger方程式

\[\begin{aligned}{} & -\frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) = i\hbar \frac{\partial\Psi(x,t)}{\partial t} \\ & -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{aligned}\]

Probability Current / 確率流

\[\begin{aligned}{} & \frac{\partial |\Psi|^2}{\partial t} = -\nabla \cdot \mathbf{j} \\ & \mathbf{j} = \frac{i\hbar}{2m} (\Psi \nabla \Psi^* - \Psi^* \nabla \Psi) \end{aligned}\]

Ehrenfest’s Theorem / Ehrenfestの定理

\[\begin{aligned}{} & \frac{d}{dt} \langle \hat{A} \rangle = -\frac{i}{\hbar} \langle [\hat{A}, \hat{H}] \rangle \\ & \frac{d}{dt} \langle \hat{x} \rangle = \left\langle \frac{\partial \hat{H}}{\partial \hat{p}} \right\rangle \\ & \frac{d}{dt} \langle \hat{p} \rangle = \left\langle -\frac{\partial \hat{H}}{\partial \hat{x}} \right\rangle \end{aligned}\]

Uncertainty Principle / 不確定性原理

\[\begin{aligned}{} & \Delta A \Delta B \geq \frac{1}{2} |\langle [A, B] \rangle| \\ & [\hat{x}, \hat{p}] = i\hbar \quad \Delta x \Delta p \geq \frac{\hbar}{2} \end{aligned}\]

1D System / 一次元系

Free Particle / 自由粒子

\[\begin{aligned}{} & V(x) = 0 \\ & \psi(x) = A e^{ikx} + B e^{-ikx} \\ & k = \frac{\sqrt{2mE}}{\hbar} \end{aligned}\]

Step Potential / 段差ポテンシャル

\[\begin{aligned}{} & V(x) = \begin{cases} 0, & x < 0 \\ V_0, & x \geq 0 \end{cases} \\ & \psi(x) = \begin{cases} A e^{ik_1 x} + B e^{-ik_1 x}, & x < 0 \\ C e^{ik_2 x}, & x \geq 0 \end{cases} \\ & k_1 = \frac{\sqrt{2mE}}{\hbar}, \quad k_2 = \frac{\sqrt{2m(E-V_0)}}{\hbar} \\ & R = \left| \frac{B}{A} \right|^2 = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2 \\ & T = \frac{k_2}{k_1} \left| \frac{C}{A} \right|^2 = \frac{4k_1 k_2}{(k_1 + k_2)^2} \end{aligned}\]

Potential Barrier / ポテンシャル障壁

\[\begin{aligned}{} & V(x) = \begin{cases} V_0, & |x| \leq \frac{a}{2} \\ 0, & |x| > \frac{a}{2} \end{cases} \\ & \psi(x) = \begin{cases} A e^{ikx} + B e^{-ikx}, & x < -\frac{a}{2} \\ C e^{\kappa x} + D e^{-\kappa x}, & -\frac{a}{2} \leq x \leq \frac{a}{2} \\ F e^{ikx}, & x > \frac{a}{2} \end{cases} \\ & k = \frac{\sqrt{2mE}}{\hbar}, \quad \kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar} \\ & R = \left| \frac{B}{A} \right|^2 = \left[ 1 + \frac{4E(V_0-E)}{V_0^2 \sinh^2(\kappa a)} \right]^{-1} \\ & T = \left| \frac{F}{A} \right|^2 = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)} \right]^{-1} \end{aligned}\]

Square Well / 井戸型ポテンシャル

\[\begin{aligned}{} & V(x) = \begin{cases} 0, & |x| < \frac{L}{2} \\ \infty, & |x| \geq \frac{L}{2} \end{cases} \\ & \psi_n(x) = \begin{cases} \sqrt{\frac{2}{L}} \cos\left(\frac{n\pi x}{L}\right), & n=1,3,5,\dots \\ \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), & n=2,4,6,\dots \end{cases} \\ & E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} \end{aligned}\]

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

\[\begin{aligned}{} & \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} + \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} && \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} - \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} \\ & \hat{a}|n\rangle = \sqrt{n}|n-1\rangle && \hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle \\ & \hat{H}|n\rangle = \hbar\omega\left( n + \frac{1}{2} \right)|n\rangle \end{aligned}\]

Symmetry / 対称性

Spatial Translational Symmetry / 空間並進対称性

\[\begin{aligned}{} & \hat{\mathbf{p}} = -i\hbar\nabla \\ & \hat{U}_{\mathbf{a}} = e^{-\frac{i}{\hbar}\mathbf{a}\cdot\hat{\mathbf{p}}} \to \hat{U}_{\mathbf{a}}\psi(\mathbf{r}) = \psi(\mathbf{r}-\mathbf{a}) \\ & \langle \hat{U}_{\mathbf{a}}\psi|\hat{H}|\hat{U}_{\mathbf{a}}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = 0 \end{aligned}\]

\[\begin{aligned}{} & \hat{U}_{\mathbf{a}} = e^{-\frac{i}{\hbar}\mathbf{a}\cdot\hat{\mathbf{p}}} \approx \hat{I} - \frac{i}{\hbar}\mathbf{a}\cdot\hat{\mathbf{p}} \\ & \hat{U}_{\mathbf{a}}\psi(\mathbf{r}) = \psi(\mathbf{r}) - \mathbf{a}\cdot\nabla\psi(\mathbf{r}) = \psi(\mathbf{r}-\mathbf{a}) \\ & \hat{U}_{\mathbf{a}}^{\dagger}\hat{H}\hat{U}_{\mathbf{a}} = \hat{H} \to [\hat{U}_{\mathbf{a}},\hat{H}] = 0 \\ & [\hat{U}_{\mathbf{a}},\hat{H}] = -\frac{i}{\hbar}\mathbf{a}\cdot[\hat{\mathbf{p}},\hat{H}] \to [\hat{\mathbf{p}},\hat{H}] = 0 \\ & \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = -\frac{i}{\hbar}\langle[\hat{\mathbf{p}},\hat{H}]\rangle \to \frac{d}{dt}\langle\hat{\mathbf{p}}\rangle = 0 \end{aligned}\]

Rotational Symmetry / 回転対称性

\[\begin{aligned}{} & \hat{\mathbf{L}} = -i\hbar\mathbf{r}\times\nabla \\ & \hat{U}_{\boldsymbol{\theta}} = e^{-\frac{i}{\hbar}\boldsymbol{\theta}\cdot\hat{\mathbf{L}}} \to \hat{U}_{\boldsymbol{\theta}}\psi(\mathbf{r}) = \psi(R_{\boldsymbol{\theta}}^{-1}\mathbf{r}) \\ & \langle \hat{U}_{\boldsymbol{\theta}}\psi|\hat{H}|\hat{U}_{\boldsymbol{\theta}}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\mathbf{L}}\rangle = 0 \end{aligned}\]

\[\begin{aligned}{} & \hat{U}_{\theta} = e^{-\frac{i}{\hbar}\theta\hat{L}_{z}} \approx \hat{I} - \frac{i}{\hbar}\theta\hat{L}_{z} \\ & \hat{U}_{\theta}\psi(\phi) = \psi(\phi) - \theta\frac{\partial\psi(\phi)}{\partial\phi} = \psi(\phi-\theta) \\ & \hat{U}_{\boldsymbol{\theta}}^{\dagger}\hat{H}\hat{U}_{\boldsymbol{\theta}} = \hat{H} \to [\hat{U}_{\boldsymbol{\theta}},\hat{H}] = 0 \\ & [\hat{U}_{\boldsymbol{\theta}},\hat{H}] = -\frac{i}{\hbar}\boldsymbol{\theta}\cdot[\hat{\mathbf{L}},\hat{H}] \to [\hat{\mathbf{L}},\hat{H}] = 0 \\ & \frac{d}{dt}\langle\hat{\mathbf{L}}\rangle = -\frac{i}{\hbar}\langle[\hat{\mathbf{L}},\hat{H}]\rangle \to \frac{d}{dt}\langle\hat{\mathbf{L}}\rangle = 0 \end{aligned}\]

Time Translational Symmetry / 時間並進対称性

\[\begin{aligned}{} & \hat{E} = i\hbar\frac{\partial}{\partial t} \\ & \hat{U}_{\tau} = e^{-\frac{i}{\hbar}\tau\hat{E}} \to \hat{U}_{\tau}\psi(t) = \psi(t+\tau) \\ & \langle \hat{U}_{\tau}\psi|\hat{H}|\hat{U}_{\tau}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{E}\rangle = 0 \end{aligned}\]

\[\begin{aligned}{} & \hat{U}_{\tau} = e^{-\frac{i}{\hbar}\tau\hat{E}} \approx \hat{I} - \frac{i}{\hbar}\tau\hat{E} \\ & \hat{U}_{\tau}\psi(t) = \psi(t) + \tau\frac{\partial\psi(t)}{\partial t} = \psi(t+\tau) \\ & \hat{U}_{\tau}^{\dagger}\hat{H}\hat{U}_{\tau} = \hat{H} \to [\hat{U}_{\tau},\hat{H}] = 0 \\ & [\hat{U}_{\tau},\hat{H}] = -\frac{i}{\hbar}\tau[\hat{E},\hat{H}] \to [\hat{E},\hat{H}] = 0 \\ & \frac{d}{dt}\langle\hat{E}\rangle = -\frac{i}{\hbar}\langle[\hat{E},\hat{H}]\rangle \to \frac{d}{dt}\langle\hat{E}\rangle = 0 \end{aligned}\]

Parity Symmetry / パリティ対称性

\[\begin{aligned}{} & \hat{\Pi}\psi(\mathbf{r}) = \psi(-\mathbf{r}), \quad \pi = \pm 1 \\ & \langle \hat{\Pi}\psi|\hat{H}|\hat{\Pi}\psi\rangle = \langle\psi|\hat{H}|\psi\rangle \to \frac{d}{dt}\langle\hat{\Pi}\rangle = 0 \end{aligned}\]

\[\begin{aligned}{} & \hat{\Pi}\psi(\mathbf{r}) = \pi\psi(\mathbf{r}) = \psi(-\mathbf{r}) \\ & \hat{\Pi}^{2}\psi(\mathbf{r}) = \pi^{2}\psi(\mathbf{r}) = \psi(\mathbf{r}) \\ & \pi^{2} = 1, \quad \pi = \pm 1 \\ & \hat{\Pi}^{\dagger}\hat{H}\hat{\Pi} = \hat{H} \to [\hat{\Pi},\hat{H}] = 0 \\ & \frac{d}{dt}\langle\hat{\Pi}\rangle = -\frac{i}{\hbar}\langle[\hat{\Pi},\hat{H}]\rangle = 0 \end{aligned}\]

Angular Momentum / 角運動量

Angular Momentum Operator / 角運動量演算子

\[\begin{aligned}{} & [\hat{J}_i, \hat{J}_j] = i\hbar \epsilon_{ijk} \hat{J}_k, \quad [\hat{\mathbf{J}}^2, \hat{J}_i] = 0 \\ & \hat{J}_+ = \hat{J}_x + i\hat{J}_y, \quad \hat{J}_- = \hat{J}_x - i\hat{J}_y \\ & \hat{J}_{\pm}|j,m\rangle = \hbar\sqrt{j(j+1) - m(m \pm 1)} |j, m \pm 1\rangle \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=0, \frac{1}{2}, 1, \dots \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \end{aligned}\]

\[\begin{aligned}{} [\hat{J}^2,\hat{J}_z] &= [\hat{J}_x^2,\hat{J}_z]+[\hat{J}_y^2,\hat{J}_z]+[\hat{J}_z^2,\hat{J}_z] \\ &= \hat{J}_x[\hat{J}_x,\hat{J}_z]+[\hat{J}_x,\hat{J}_z]\hat{J}_x+\hat{J}_y[\hat{J}_y,\hat{J}_z]+[\hat{J}_y,\hat{J}_z]\hat{J}_y \\ &= \hat{J}_x(-i\hbar\hat{J}_y)+(-i\hbar\hat{J}_y)\hat{J}_x+\hat{J}_y(i\hbar\hat{J}_x)+(i\hbar\hat{J}_x)\hat{J}_y \\ &= 0 \end{aligned} \\ \hat{J}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle,\quad \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle \\ \begin{aligned}{} [\hat{J}^2,\hat{J}_{\pm}] &= [\hat{J}^2,\hat{J}_x]\pm i[\hat{J}^2,\hat{J}_y] = 0 \end{aligned} \\ \begin{aligned}{} \hat{J}^2\hat{J}_{\pm}|j,m\rangle &= \hat{J}_{\pm}\hat{J}^2|j,m\rangle = \hbar^2 j(j+1)\hat{J}_{\pm}|j,m\rangle \end{aligned} \\ \begin{aligned}{} [\hat{J}_z,\hat{J}_{\pm}] &= [\hat{J}_z,\hat{J}_x]\pm i[\hat{J}_z,\hat{J}_y] \\ &= i\hbar\hat{J}_y \pm i(-i\hbar\hat{J}_x) \\ &= \pm \hbar\hat{J}_{\pm} \end{aligned} \\ \begin{aligned}{} \hat{J}_z\hat{J}_{\pm}|j,m\rangle &= \hat{J}_{\pm}\hat{J}_z|j,m\rangle \pm \hbar\hat{J}_{\pm}|j,m\rangle \\ &= \hbar(m\pm 1)\hat{J}_{\pm}|j,m\rangle \end{aligned} \\ \hat{J}_{\pm}|j,m\rangle = C_{\pm}|j,m\pm 1\rangle \\ \begin{aligned}{} |C_{\pm}|^2 &= \langle j,m|\hat{J}_{\pm}^{\dagger}\hat{J}_{\pm}|j,m\rangle \\ &= \langle j,m|\hat{J}_x^2+\hat{J}_y^2\pm i[\hat{J}_x,\hat{J}_y]|j,m\rangle \\ &= \langle j,m|\hat{J}^2-\hat{J}_z^2\mp\hbar\hat{J}_z|j,m\rangle \\ &= \hbar^2 j(j+1)-\hbar^2 m^2 \mp \hbar^2 m \end{aligned} \\ C_{\pm} = \hbar\sqrt{j(j+1)-m(m\pm 1)} \\ \langle j,m|\hat{J}^2-\hat{J}_z^2|j,m\rangle = \langle j,m|\hat{J}_x^2+\hat{J}_y^2|j,m\rangle \geq 0 \\ \hbar^2[j(j+1)-m^2]\geq 0,\quad m_{\min}\leq m\leq m_{\max} \\ \hat{J}_+|j,m_{\max}\rangle = 0,\quad j(j+1)-m_{\max}(m_{\max}+1)=0 \\ \hat{J}_-|j,m_{\min}\rangle = 0,\quad j(j+1)-m_{\min}(m_{\min}-1)=0 \\ m_{\max}=j,\quad m_{\min}=-j,\quad m=-j,-j+1,\ldots,j \\ j-(-j)\in\mathbb{Z}_{\geq 0},\quad j=0,\frac{1}{2},1,\ldots\]

Orbital Angular Momentum / 軌道角運動量

\[\begin{aligned}{} & \hat{\mathbf{L}} = -i\hbar \mathbf{r} \times \nabla \to [\hat{L}_i, \hat{L}_j] = i\hbar \epsilon_{ijk} \hat{L}_k\\ & \hat{\mathbf{L}}^2 = -\hbar^2 \left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] \\ & \hat{L}_z = -i\hbar\frac{\partial}{\partial\phi} \\ & \hat{\mathbf{L}}^2|l,m\rangle = \hbar^2 l(l+1)|l,m\rangle, \quad l=0, 1, 2, \dots \\ & \hat{L}_z|l,m\rangle = \hbar m|l,m\rangle, \quad m=-l, -l+1, \dots, l \\ & Y_l^m(\theta, \phi) = (-1)^m \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi} \end{aligned}\]

\[\begin{aligned}{} [\hat{L}_x,\hat{L}_y] &= -\hbar^2\left(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\right)=i\hbar\hat{L}_z \\ [\hat{L}_y,\hat{L}_z] &= -\hbar^2\left(z\frac{\partial}{\partial y}-y\frac{\partial}{\partial z}\right)=i\hbar\hat{L}_x \\ [\hat{L}_z,\hat{L}_x] &= -\hbar^2\left(x\frac{\partial}{\partial z}-z\frac{\partial}{\partial x}\right)=i\hbar\hat{L}_y \end{aligned} \\ \begin{aligned}{} \hat{L}^2 &= -\hbar^2\left(\mathbf{r}\times\nabla\right)^2 \\ &= -\hbar^2\left[r^2\nabla^2-\left(\mathbf{r}\cdot\nabla\right)^2-\mathbf{r}\cdot\nabla\right] \\ &= -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] \end{aligned} \\ \hat{L}_z = -i\hbar\left(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)=-i\hbar\frac{\partial}{\partial\phi} \\ \hat{L}_zY_l^m = -i\hbar\frac{\partial Y_l^m}{\partial\phi}=\hbar mY_l^m \rightarrow Y_l^m(\theta,\phi)=\Theta_l^m(\theta)e^{im\phi} \\ Y_l^m(\theta,\phi+2\pi) = Y_l^m(\theta,\phi) \rightarrow e^{i2\pi m}=1,\quad m\in\mathbb{Z} \\ \begin{aligned}{} \hat{L}^2Y_l^m &= -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Theta_l^m}{\partial\theta}\right)e^{im\phi}-\frac{m^2}{\sin^2\theta}\Theta_l^me^{im\phi}\right] \\ &= \hbar^2l(l+1)\Theta_l^me^{im\phi} \end{aligned} \\ \frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta_l^m}{d\theta}\right)+\left[l(l+1)-\frac{m^2}{\sin^2\theta}\right]\Theta_l^m = 0 \\ x = \cos\theta,\quad \frac{d}{d\theta}=-\sin\theta\frac{d}{dx} \\ (1-x^2)\frac{d^2\Theta_l^m}{dx^2}-2x\frac{d\Theta_l^m}{dx}+\left[l(l+1)-\frac{m^2}{1-x^2}\right]\Theta_l^m = 0 \\ \Theta_l^m(x) = AP_l^m(x) \rightarrow Y_l^m(\theta,\phi)=AP_l^m(\cos\theta)e^{im\phi}\]

Spin Angular Momentum / スピン角運動量

\[\begin{aligned}{} & \hat{\mathbf{S}} = \frac{\hbar}{2} \boldsymbol{\sigma} \to [\hat{S}_i, \hat{S}_j] = i\hbar \epsilon_{ijk} \hat{S}_k\\ & \hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \\ & \hat{\mathbf{S}}^2|s,m\rangle = \hbar^2 s(s+1)|s,m\rangle, \quad s=\frac{1}{2} \\ & \hat{S}_z|s,m\rangle = \hbar m|s,m\rangle, \quad m=-\frac{1}{2}, \frac{1}{2} \\ & \hat{H} = -\gamma \hat{\mathbf{S}} \cdot \mathbf{B}, \quad \omega_L = \gamma B \end{aligned}\]

\[\begin{aligned}{} [\hat{S}_x,\hat{S}_y] &= \frac{\hbar^2}{4}\left[\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}-\begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}\right] = i\hbar\hat{S}_z \\ [\hat{S}_y,\hat{S}_z] &= \frac{\hbar^2}{4}\left[\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}-\begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}\right] = i\hbar\hat{S}_x \\ [\hat{S}_z,\hat{S}_x] &= \frac{\hbar^2}{4}\left[\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}-\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right] = i\hbar\hat{S}_y \end{aligned} \\ \begin{aligned}{} \hat{S}_z \begin{pmatrix} 1 \\ 0 \end{pmatrix} &= \frac{\hbar}{2}\begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad m=\frac{1}{2} \\ \hat{S}_z \begin{pmatrix} 0 \\ 1 \end{pmatrix} &= -\frac{\hbar}{2}\begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad m=-\frac{1}{2} \end{aligned} \\ \hat{H}=-\gamma\hat{\mathbf{S}}\cdot\mathbf{B}, \quad \mathbf{B} = B\hat{\mathbf{z}} \rightarrow \hat{H}=-\gamma\hat{S}_zB \\ \hat{H}|\!\uparrow\rangle=-\frac{\hbar\gamma B}{2}|\!\uparrow\rangle, \quad \hat{H}|\!\downarrow\rangle=\frac{\hbar\gamma B}{2}|\!\downarrow\rangle \\ \begin{aligned}{} |\psi(t)\rangle &= e^{-\frac{i}{\hbar}\hat{H}t}\left(a|\!\uparrow\rangle+b|\!\downarrow\rangle\right) \\ &= ae^{i\frac{\gamma B}{2}t}|\!\uparrow\rangle+be^{-i\frac{\gamma B}{2}t}|\!\downarrow\rangle \\ &= e^{i\frac{\gamma B}{2}t}\left(a|\!\uparrow\rangle+be^{-i\gamma Bt}|\!\downarrow\rangle\right) \\ \end{aligned} \\ \begin{aligned}{} \langle S_x\rangle &= \langle\psi(t)|\hat{S}_x|\psi(t)\rangle = \hbar |ab|\cos(\gamma Bt+\phi) \\ \langle S_y\rangle &= \langle\psi(t)|\hat{S}_y|\psi(t)\rangle = -\hbar |ab|\sin(\gamma Bt+\phi) \\ \langle S_z\rangle &= \langle\psi(t)|\hat{S}_z|\psi(t)\rangle = \frac{\hbar}{2}\left(|a|^2-|b|^2\right) \end{aligned}\]

Addition of Angular Momentum / 角運動量の合成

\[\begin{aligned}{} & \hat{\mathbf{J}} = \hat{\mathbf{J}}_1 + \hat{\mathbf{J}}_2 \to [\hat{J}_i, \hat{J}_j] = i\hbar \epsilon_{ijk} \hat{J}_k \\ & \hat{\mathbf{J}}^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle, \quad j=|j_1-j_2|, |j_1-j_2|+1, \dots, j_1+j_2 \\ & \hat{J}_z|j,m\rangle = \hbar m|j,m\rangle, \quad m=-j, -j+1, \dots, j \\ & |j_1,j_2;j,m\rangle = \sum_{m_1,m_2} C_{j_1m_1j_2m_2}^{jm} |j_1,m_1\rangle |j_2,m_2\rangle \\ & C_{j_1m_1j_2m_2}^{jm} \neq 0 \to m = m_1 + m_2 \\ & \sum_{m_1,m_2} C_{j_1m_1j_2m_2}^{jm} C_{j_1m_1j_2m_2}^{j'm'} = \delta_{jj'}\delta_{mm'} \end{aligned}\]

\[\begin{aligned}{} [\hat{J}_i,\hat{J}_j] &=[\hat{J}_{1i},\hat{J}_{1j}]+[\hat{J}_{1i},\hat{J}_{2j}]+[\hat{J}_{2i},\hat{J}_{1j}]+[\hat{J}_{2i},\hat{J}_{2j}] \\ &=i\hbar \hat{J}_{1k}+0+0+i\hbar \hat{J}_{2k} =i\hbar \hat{J}_k \end{aligned} \\ \hat{J}^2|j,m\rangle=\hbar^2j(j+1)|j,m\rangle,\quad \hat{J}_z|j,m\rangle=\hbar m|j,m\rangle \\ \hat{J}_z|j_1,m_1\rangle|j_2,m_2\rangle=\hbar(m_1+m_2)|j_1,m_1\rangle|j_2,m_2\rangle \\ m_{\max}=m_{1\max}+m_{2\max}=j_1+j_2 \\ \sum_{m_{\min}}^{m_{\max}}(2j+1)=(2j_1+1)(2j_2+1) \\ m_{\min}=\sqrt{j_1^2+j_2^2-2j_1j_2}=|j_1-j_2| \\ \begin{aligned}{} \hat{J}_z|j_1,j_2;j,m\rangle &=\hbar m|j_1,j_2;j,m\rangle \\ &=\hbar m\sum_{m_1,m_2}C^{jm}_{j_1m_1j_2m_2}|j_1,m_1\rangle|j_2,m_2\rangle \\ \hat{J}_z|j_1,j_2;j,m\rangle&=(\hat{J}_{1z}+\hat{J}_{2z})|j_1,j_2;j,m\rangle \\ &=\sum_{m_1,m_2}C^{jm}_{j_1m_1j_2m_2}(\hbar m_1+\hbar m_2)|j_1,m_1\rangle|j_2,m_2\rangle \end{aligned} \\ \sum_{m_1,m_2}\hbar(m-m_1-m_2)C^{jm}_{j_1m_1j_2m_2}|j_1,m_1\rangle|j_2,m_2\rangle=0 \\ C^{jm}_{j_1m_1j_2m_2}\neq0\rightarrow m = m_1-m_2 \\ \begin{aligned}{} \langle j_1,j_2;j,m|j_1,j_2;j',m'\rangle &=\sum_{m_1,m_2}\sum_{m_1',m_2'}C^{jm}_{j_1m_1j_2m_2}C^{j'm'}_{j_1m_1'j_2m_2'}\delta_{m_1,m_1'}\delta_{m_2,m_2'} \\ &=\sum_{m_1,m_2}C^{jm}_{j_1m_1j_2m_2}C^{j'm'}_{j_1m_1j_2m_2}\quad (C\in\mathbb{R}) \\ &=\delta_{jj'}\delta_{mm'} \end{aligned}\]

3D System / 三次元系

Central Potential / 中心力ポテンシャル

\[\begin{aligned}{} & -\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r}) \\ & V(\mathbf{r}) = V(r), \quad \psi(\mathbf{r}) = R(r)Y_l^m(\theta,\phi), \quad u(r) = rR(r) \\ & -\frac{\hbar^2}{2m}\frac{d^2u(r)}{dr^2} + \left[V(r) + \frac{\hbar^2 l(l+1)}{2mr^2}\right]u(r) = Eu(r) \end{aligned}\]

\[\begin{aligned}{} \nabla^2 &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] \\ &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) - \frac{\hat{L}^2}{\hbar^2 r^2} \end{aligned} \\ V(\mathbf{r})=V(r) \rightarrow [\hat{H}, \hat{L}^2]=0, \; [\hat{H}, \hat{L}_z]=0 \rightarrow \psi=R(r)Y_l^m(\theta, \phi) \\ \begin{aligned}{} & -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi \\ = &-\frac{\hbar^2}{2m}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial\psi}{\partial r}\right) - \frac{\hat{L}^2}{\hbar^2 r^2}\psi\right] + V\psi \\ = &-\frac{\hbar^2}{2m}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)Y_l^m - \frac{R}{\hbar^2 r^2}\hat{L}^2Y_l^m\right] + VRY_l^m \\ = &-\frac{\hbar^2}{2m}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)Y_l^m - \frac{R}{\hbar^2 r^2}\hbar^2 l(l+1)Y_l^m\right] + VRY_l^m \\ = &ERY_l^m \end{aligned} \\ \begin{aligned}{} & -\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \frac{\hbar^2 l(l+1)}{2mr^2}R + VR \\ = & -\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left[r^2\frac{d}{dr}\left(\frac{u}{r}\right)\right] + \frac{\hbar^2 l(l+1)}{2mr^2}\frac{u}{r} + V\frac{u}{r} \\ = & -\frac{\hbar^2}{2m}\frac{1}{r}\frac{d^2u}{dr^2} + \frac{\hbar^2 l(l+1)}{2mr^2}\frac{u}{r} + V\frac{u}{r} = E\frac{u}{r} \end{aligned}\]

Infinite Spherical Well / 無限球形井戸

\[\begin{aligned}{} & V(r) = \begin{cases} 0, & r < a \\ \infty, & r \geq a \end{cases} \\ & \psi_{nlm}(r,\theta,\phi) = \frac{\sqrt{2}}{a^{3/2}\left|j_{l+1}(\alpha_{nl})\right|} j_l(k_{nl}r)Y_l^m(\theta,\phi) \\ & j_l(\alpha_{nl})=0, \quad k_{nl} = \frac{\alpha_{nl}}{a} \\ & E_{nl} = \frac{\hbar^2 k_{nl}^2}{2m}, \quad n = 1,2,3,\dots, \quad l = 0,1,2,\dots \end{aligned}\]

\[-\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \frac{\hbar^2 l(l+1)}{2mr^2}R = ER \\ \frac{d^2R}{dr^2} + \frac{2}{r}\frac{dR}{dr} + \left[\frac{2mE}{\hbar^2} - \frac{l(l+1)}{r^2}\right]R = 0 \\ k = \frac{\sqrt{2mE}}{\hbar}, \quad E = \frac{\hbar^2 k^2}{2m}, \quad x = kr \\ x^2\frac{d^2R}{dx^2} + 2x\frac{dR}{dx} + \left[x^2 - l(l+1)\right]R = 0 \\ R = A j_l(x) = A j_l(kr), \quad \psi = A j_l(kr)Y_l^m(\theta,\phi) \\ \psi(a,\theta,\phi) = A j_l(ka)Y_l^m(\theta,\phi) = 0, \quad ka = \alpha_{nl}\]

Isotropic Harmonic Oscillator / 等方性調和振動子

\[\begin{aligned}{} & V(r) = \frac{1}{2}m\omega^2 r^2 \\ & \psi_{n_x n_y n_z}(x,y,z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ & N = n_x+n_y+n_z, \quad g_N = \frac{(N+1)(N+2)}{2} \\ & E_N = \hbar\omega\left(N+\frac{3}{2}\right), \quad N = 0,1,2,\dots \end{aligned}\]

\[-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)\psi + \frac{1}{2}m\omega^2(x^2+y^2+z^2)\psi = E\psi \\ (\hat{H}_x+\hat{H}_y+\hat{H}_z)X(x)Y(y)Z(z) = EX(x)Y(y)Z(z) \\ \frac{\hat{H}_xX}{X} + \frac{\hat{H}_yY}{Y} + \frac{\hat{H}_zZ}{Z} = E = E_x+E_y+E_z \\ \hat{H}_x\psi_{n_x} = E_{n_x}\psi_{n_x}, \quad \hat{H}_y\psi_{n_y} = E_{n_y}\psi_{n_y}, \quad \hat{H}_z\psi_{n_z} = E_{n_z}\psi_{n_z} \\ \psi_{n_x n_y n_z}(x, y, z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_{n_z}(z) \\ \begin{aligned}{} E &= E_{n_x}+E_{n_y}+E_{n_z} \\ &= \hbar\omega\left(n_x+\frac{1}{2}\right) + \hbar\omega\left(n_y+\frac{1}{2}\right) + \hbar\omega\left(n_z+\frac{1}{2}\right) \\ &= \hbar\omega\left(N+\frac{3}{2}\right) \end{aligned} \\ N = n_x+n_y+n_z, \quad g_N = \frac{(N+2)!}{N!\,2!} = \frac{(N+1)(N+2)}{2}\]

Hydrogen Atom / 水素原子

\[\begin{aligned}{} & V(r) = -\frac{e^2}{4\pi\epsilon_0 r} \\ & \psi_{nlm}(r,\theta,\phi) = \sqrt{\left(\frac{2}{na_0}\right)^3 \frac{(n-l-1)!}{2n[(n+l)!]}} e^{-\rho} (2\rho)^l L_{n-l-1}^{2l+1}(2\rho) Y_l^m(\theta,\phi) \\ & a_0 = \frac{4\pi\epsilon_0\hbar^2}{me^2}, \quad \rho = \frac{r}{na_0} \\ & E_n = -\frac{me^4}{2(4\pi\varepsilon_0)^2\hbar^2}\frac{1}{n^2}, \quad n=1,2,3,\dots \end{aligned}\]

\[-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + \left[-\frac{e^2}{4\pi\varepsilon_0 r} + \frac{\hbar^2 l(l+1)}{2mr^2}\right]u = Eu \\ a_0 = \frac{4\pi\varepsilon_0\hbar^2}{me^2}, \quad \kappa^2 = -\frac{2mE}{\hbar^2}, \quad \rho = \kappa r \\ \frac{d^2u}{dr^2} + \left[\frac{2}{a_0r} - \frac{l(l+1)}{r^2} - \kappa^2\right]u = 0 \\ \begin{aligned}{} & r \to 0: && \frac{d^2u}{dr^2} \simeq \frac{l(l+1)}{r^2}u, && u \sim r^{l+1} \\ & r \to \infty: && \frac{d^2u}{dr^2} \simeq \kappa^2u, && u \sim e^{-\kappa r} \end{aligned} \\ R(r) = \frac{u(r)}{r} = e^{-\rho}\rho^l v(\rho) \\ -\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \left[-\frac{e^2}{4\pi\varepsilon_0 r} + \frac{\hbar^2 l(l+1)}{2mr^2}\right]R = ER \\ \frac{d^2R}{dr^2} + \frac{2}{r}\frac{dR}{dr} + \left[\frac{2}{a_0r} - \frac{l(l+1)}{r^2} - \kappa^2\right]R = 0 \\ \rho\frac{d^2v}{d\rho^2} + (2l+2-2\rho)\frac{dv}{d\rho} + \left(\frac{2}{a_0\kappa} -2l-2\right)v = 0 \\ x\frac{d^2v}{dx^2} + (2l+2-x)\frac{dv}{dx} + \left(\frac{1}{a_0\kappa} -l-1\right)v = 0 \\ \frac{1}{a_0\kappa}=n, \quad n=1,2,3,\cdots \\ n-l-1\geq 0, \quad l=0,1,\cdots,n-1 \\ \kappa = \frac{1}{na_0}, \quad \rho=\frac{r}{na_0} \\ v = L_{n-l-1}^{2l+1}(x) = L_{n-l-1}^{2l+1}(2\rho) \\ \psi = Ae^{-\rho}\rho^l L_{n-l-1}^{2l+1}(2\rho) Y_l^m(\theta,\phi) \\ E = -\frac{\hbar^2}{2ma_0^2}\frac{1}{n^2} = -\frac{me^4}{2(4\pi\varepsilon_0)^2\hbar^2}\frac{1}{n^2}\]

Electromagnetic Potential / 電磁ポテンシャル

\[\begin{aligned}{} & \hat{H}\psi = \left[\frac{1}{2m}(\hat{\mathbf{p}} - q\mathbf{A})^2 + q\Phi\right]\psi = i\hbar\frac{\partial\psi}{\partial t} \\ & \mathbf{A}' = \mathbf{A} + \nabla\chi, \quad \Phi' = \Phi - \frac{\partial\chi}{\partial t}, \quad \psi' = e^{\frac{i}{\hbar}q\chi}\psi \\ & \hat{H'}\psi' = \left[\frac{1}{2m}(\hat{\mathbf{p}} - q\mathbf{A'})^2 + q\Phi'\right]\psi' = i\hbar\frac{\partial\psi'}{\partial t} \\ \end{aligned}\]

\[\begin{aligned}{} (\hat{\mathbf{p}}-q\mathbf{A}')\psi' &= \left( -i\hbar\nabla -q\mathbf{A} -q\nabla\chi \right) e^{\frac{i}{\hbar}q\chi}\psi \\ &= e^{\frac{i}{\hbar}q\chi} \left( -i\hbar\nabla -q\mathbf{A} \right)\psi \\ (\hat{\mathbf{p}}-q\mathbf{A}')^2\psi' &= e^{\frac{i}{\hbar}q\chi} \left( -i\hbar\nabla -q\mathbf{A} \right)^2\psi \\ \frac{(\hat{\mathbf{p}}-q\mathbf{A}')^2}{2m}\psi' &= e^{\frac{i}{\hbar}q\chi} \frac{(\hat{\mathbf{p}}-q\mathbf{A})^2}{2m}\psi \\ q\Phi'\psi' &= q\left( \Phi-\frac{\partial\chi}{\partial t} \right) e^{\frac{i}{\hbar}q\chi}\psi \\ &= e^{\frac{i}{\hbar}q\chi}q\Phi\psi - q\frac{\partial\chi}{\partial t} e^{\frac{i}{\hbar}q\chi}\psi \\ i\hbar\frac{\partial\psi'}{\partial t} &= i\hbar\frac{\partial}{\partial t} \left( e^{\frac{i}{\hbar}q\chi}\psi \right) \\ &= e^{\frac{i}{\hbar}q\chi} i\hbar\frac{\partial\psi}{\partial t} - q\frac{\partial\chi}{\partial t} e^{\frac{i}{\hbar}q\chi}\psi \\ \hat{H}'\psi' - i\hbar\frac{\partial\psi'}{\partial t} &= e^{\frac{i}{\hbar}q\chi} \left( \hat{H}\psi - i\hbar\frac{\partial\psi}{\partial t} \right) =0 \end{aligned}\]

Charged Particle / 荷電粒子

\[\begin{aligned}{} & E_{n,k_z} = \hbar\omega_c\left(n+\frac{1}{2}\right) + \frac{\hbar^2 k_z^2}{2m}, \quad \omega_c=\frac{|q|B}{m} \\ & \psi = \psi_0 \exp\left(\frac{i}{\hbar}q\int_C \mathbf{A}\cdot d\mathbf{l}\right),\quad \Delta\varphi = \frac{q}{\hbar}\int_S \mathbf{B}\cdot d\mathbf{S} \end{aligned}\]

\[\mathbf{B}=B\hat{\mathbf{z}}, \quad \nabla\times\mathbf{A}=\mathbf{B} \to \mathbf{A}=Bx\hat{\mathbf{y}} \\ \hat{H} = \frac{1}{2m} \left[ \hat{p}_x^2 + (\hat{p}_y-qBx)^2 + \hat{p}_z^2 \right] \\ [\hat{H},\hat{p}_y]=0, \quad [\hat{H},\hat{p}_z]=0 \to \psi=e^{ik_y y}e^{ik_z z}X(x) \\ \begin{aligned}{} \hat{H}\psi &= \frac{1}{2m} \left[ \hat{p}_x^2 + (\hbar k_y-qBx)^2 + (\hbar k_z)^2 \right]\psi \\ &= \left[ \frac{\hat{p}_x^2}{2m} + \frac{q^2B^2}{2m} \left(x-\frac{\hbar k_y}{qB}\right)^2 + \frac{\hbar^2k_z^2}{2m} \right]\psi \\ &= \left[ \frac{\hat{p}_x^2}{2m} + \frac{m\omega_c^2}{2}(x-x_0)^2 + \frac{\hbar^2k_z^2}{2m} \right]\psi \end{aligned} \\ \omega_c = \frac{|q|B}{m}, \quad E = \hbar\omega_c\left(n+\frac{1}{2}\right) + \frac{\hbar^2k_z^2}{2m} \\ \mathbf{A}_0 = \mathbf{A}+\nabla\chi=0 \to \mathbf{A}=-\nabla\chi, \quad \chi=-\int_{\mathbf{r}_0}^{\mathbf{r}}\mathbf{A}\cdot d\mathbf{l} \\ \psi_0 = \psi e^{\frac{i}{\hbar}q\chi} \to \psi = \psi_0 e^{-\frac{i}{\hbar}q\chi} = \psi_0 \exp\left( \frac{i}{\hbar}q\int_C\mathbf{A}\cdot d\mathbf{l} \right) \\ \begin{aligned}{} \Delta\varphi &= \frac{q}{\hbar}\int_{C_1}\mathbf{A}\cdot d\mathbf{l} - \frac{q}{\hbar}\int_{C_2}\mathbf{A}\cdot d\mathbf{l} \\ &= \frac{q}{\hbar}\oint_C\mathbf{A}\cdot d\mathbf{l} = \frac{q}{\hbar}\int_S(\nabla\times\mathbf{A})\cdot d\mathbf{S} \\ &= \frac{q}{\hbar}\int_S\mathbf{B}\cdot d\mathbf{S} = \frac{q\Phi_B}{\hbar} \end{aligned}\]

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

\[\begin{aligned}{} & j^\mu = c\psi^\dagger\gamma^0\gamma^\mu\psi \\ & \partial_\mu j^\mu = 0 \end{aligned}\]

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Classical Mechanics / 古典力学

Newtonian Mechanics / Newton力学

Equation of Motion / 運動方程式

Momentum / 運動量

Angular Momentum / 角運動量

Energy / エネルギー

Lagrangian Mechanics / Lagrange力学

Lagrange’s Equation / Langrange方程式

Conservation Law / 保存則

Galilean Transformation / ガリレイ変換

Virial Theorem / Virial定理

Hamiltonian Mechanics / Hamilton力学

Hamilton’s Equation / Hamilton方程式

Poisson Bracket / Poisson括弧

Canonical Transformation / 正準変換

Hamilton–Jacobi Equation / Hamilton–Jacobi方程式

Liouville’s Theorem / Liouvilleの定理

Oscillation / 振動

Small Oscillation / 微小振動

Damped Oscillation / 減衰振動

Forced Oscillation / 強制振動

Coupled Oscillation / 連成振動

Wave / 波動

Wave Equation / 波動方程式

Bounded Wave / 有界波

Free Wave / 自由波

Central Force Motion / 中心力運動

Reduced Mass / 換算質量

Orbit / 軌道

Kepler Problem / Kepler問題

Collision / 衝突

Scattering / 散乱

Gravity / 重力

Rigid Body Motion / 剛体運動

Non-Inertial Frame / 非慣性系

Euler Angle / Euler角

Inertia Tensor / 慣性テンソル

Euler’s Equation / Eulerの運動方程式

Symmetrical Top (Free) / 対称コマ（自由）

Symmetrical Top (Gravity) / 対称コマ（重力）

Asymmetrical Top (Free) / 非対称コマ（自由）

Continuum Mechanics / 連続体力学

Fundamental Equation / 基礎方程式

Elastic Body / 弾性体

Fluid / 流体

Bernoulli’s Principle / Bernoulliの定理

Special Relativity / 特殊相対性理論

Lorentz Transformation / Lorentz変換

Consequence of Transformation / 変換の帰結

Relativistic Dynamics / 相対論的力学

Four-Vector / ４元ベクトル

Electromagnetism / 電磁気学

Maxwell’s Equation / Maxwell方程式

Charge and Current / 電荷と電流

Electromagnetic Field / 電磁場

Electromagnetic Potential / 電磁ポテンシャル

Energy Conservation / エネルギー保存

Momentum Conservation / 運動量保存

Electrostatics / 静電場

Electric Field / 電場

Electric Potential / 電位

Electric Energy / 電気エネルギー

Electric Dipole / 電気双極子

Uniqueness Theorem / 一意性定理

Capacitor / コンデンサー

Magnetostatics / 静磁場

Magnetic Field / 磁場

Magnetic Potential / 磁位

Magnetic Energy / 磁気エネルギー

Magnetic Dipole / 磁気双極子

Electromagnetic Induction / 電磁誘導

Inductor / インダクタ

Electromagnetism in Matter / 物質中の電磁気学

Polarization / 分極

Electric Displacement / 電束密度

Magnetization / 磁化

Auxiliary Field / 補助磁場

Maxwell’s Equations in Matter / 物質中のMaxwell方程式

Boundary Conditions / 境界条件