Equation
Published:
Some random equations
Classical Mechanics / 古典力学
\[\begin{aligned} & S = \int_{t_1}^{t_2} L\left(q_i,\dot q_i,t\right)\,dt \\ & \delta S = 0 \end{aligned}\]
Electromagnetism / 電磁気学
\[\begin{aligned} &\nabla\cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\ &\nabla\times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla\cdot \mathbf{B} = 0 \\ &\nabla\times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}\]
Statistical Mechanics / 統計力学
\[\begin{aligned} & S = - k_B \sum_i p_i \ln p_i \\ & \delta S = 0 \\ & dU = T\,dS - P\,dV + \mu\,dN \end{aligned}\]
Quantum Mechanics / 量子力学
\[\begin{aligned} & |\psi\rangle \in \mathcal{H} \\ & \hat{A} = \hat{A}^\dagger \\ & P(a_n) = |\langle a_n | \psi \rangle|^2 \\ & i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H}|\psi(t)\rangle \end{aligned}\]
Formulation / 定式化
State / 状態
\[\begin{aligned} & \langle m | n \rangle = \delta_{mn} && \langle a | a' \rangle = \delta(a - a') \\ & \sum_n |n\rangle\langle n| = \mathbb I && \int |a\rangle\langle a| \, da = \mathbb I \\ & |\psi\rangle = \sum_n |n\rangle\langle n|\psi\rangle && |\psi\rangle = \int |a\rangle\langle a|\psi\rangle \, da \end{aligned}\]
Observable / オブザーバブル
\[\begin{aligned} & \hat{A}|a\rangle = a|a\rangle \quad \langle a | a \rangle = \mathbb I \\ & a = a^* \quad \langle a_n | a_m \rangle = \delta_{nm} \\ & \hat{A} = \sum_a a |a\rangle\langle a| \quad \hat{A} = \int a |a\rangle\langle a| \, da \end{aligned}\]
Measurement / 測定
\[\begin{aligned} & p(a) = \langle \psi | \hat{P}_a | \psi \rangle \\ & \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle \\ & |\psi\rangle \to \frac{\hat{P}_a |\psi\rangle}{\sqrt{\langle \psi | \hat{P}_a | \psi \rangle}} \end{aligned}\]
Evolution / 発展
\[\begin{aligned} & |\psi(t)\rangle = \hat U(t,t_0)\,|\psi(t_0)\rangle \\ & \hat U^\dagger(t,t_0)\,\hat U(t,t_0) = \mathbb I \\ & \hat U(t,t_0) = \exp\left[-\frac{i}{\hbar}\,\hat H\,\bigl(t-t_0\bigr)\right] \\ & \hat U(t,t_0) = \mathcal T \exp\left[-\frac{i}{\hbar}\int_{t_0}^{t}\hat H(t')\,dt'\right] \\ \end{aligned}\]
Wave Mechanics / 波動力学
Wave function / 波動関数
\[\begin{aligned} & \langle x | \psi \rangle = \psi(x) \\ & \langle \phi | \psi \rangle = \int_{-\infty}^{\infty} \phi^*(x) \psi(x) dx \\ & \langle x | \hat{x} | \psi \rangle = x \psi(x) \\ & \langle x | \hat{p} | \psi \rangle = -i\hbar \frac{\partial}{\partial x} \psi(x) \end{aligned}\]
Schrödinger equation / シュレーディンガー方程式
\[\begin{aligned} & -\frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t) = i\hbar \frac{\partial\Psi(x,t)}{\partial t} \\ & -\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{aligned}\]
Change of basis / 基底変換
\[\begin{aligned} & \langle x | x' \rangle = \delta(x - x') && \langle p | p' \rangle = \delta(p - p') \\ & \langle x | p \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{\frac{i}{\hbar}px} && \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-\frac{i}{\hbar}px} \\ & \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-\frac{i}{\hbar}px} dx && \psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{\frac{i}{\hbar}px} dp \end{aligned}\]
Uncertainty principle / 不確定性原理
\[\begin{aligned} & [\hat{x}, \hat{p}] = i\hbar \\ & \Delta x \Delta p \geq \frac{\hbar}{2} \end{aligned} \\ \psi(x) = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp\left[-\frac{(x-x_0)^2}{4\sigma^2}\right] \exp\left[\frac{i}{\hbar}p_0(x-x_0)\right]\]
Probability current / 確率流密度
\[\begin{aligned} & \frac{\partial |\Psi|^2}{\partial t} = -\nabla \cdot \mathbf{j} \\ & \mathbf{j} = \frac{\hbar}{2mi} (\Psi^* \nabla \Psi - \Psi \nabla \Psi^*) \end{aligned}\]
Ehrenfest’s theorem / エーレンフェストの定理
\[\begin{aligned} & \frac{d}{dt} \langle \hat{A} \rangle = \frac{1}{i\hbar} \langle [\hat{A}, \hat{H}] \rangle \\ & \frac{d}{dt} \langle \hat{x} \rangle = \left\langle \frac{\partial \hat{H}}{\partial \hat{p}} \right\rangle \\ & \frac{d}{dt} \langle \hat{p} \rangle = \left\langle -\frac{\partial \hat{H}}{\partial \hat{x}} \right\rangle \end{aligned}\]
1D System / １次元系
Free Particle / 自由粒子
\[\begin{aligned} & V(x) = 0 \\ & \psi_k(x) = \frac{1}{\sqrt{2\pi}} e^{ikx} \\ & E = \frac{\hbar^2 k^2}{2m} \end{aligned}\]
Step Potential / 段差ポテンシャル
\[\begin{aligned} & V(x) = \begin{cases} 0, & x < 0 \\ V_0, & x \geq 0 \end{cases} \\ & \psi_{k_1}(x) = \begin{cases} \frac{1}{\sqrt{2\pi}} \left( e^{ik_1 x} + \frac{k_1 - k_2}{k_1 + k_2} e^{-ik_1 x} \right), & x < 0 \\ \frac{1}{\sqrt{2\pi}} \frac{2k_1}{k_1 + k_2} e^{ik_2 x}, & x \geq 0 \end{cases} \\ & E = \frac{\hbar^2 k_1^2}{2m} = \frac{\hbar^2 k_2^2}{2m} + V_0 \end{aligned}\]
Square Well / 井戸型ポテンシャル
\[\begin{aligned} & V(x) = \begin{cases} 0, & |x| < \frac{L}{2} \\ \infty, & |x| \geq \frac{L}{2} \end{cases} \\ & \psi_n(x) = \begin{cases} \sqrt{\frac{2}{L}} \cos\left(\frac{n\pi x}{L}\right), & n=1,3,5,\dots \\ \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), & n=2,4,6,\dots \end{cases} \\ & E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} \end{aligned}\]
Harmonic Oscillator / 調和振動子
\[\begin{aligned} & V(x) = \frac{1}{2} m \omega^2 x^2 \\ & \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{m\omega}{\pi\hbar} \right)^{1/4} H_n \left( \sqrt{\frac{m\omega}{\hbar}} x \right) \exp\left( -\frac{m\omega x^2}{2\hbar} \right) \\ & E_n = \hbar \omega \left( n + \frac{1}{2} \right) \end{aligned}\]
Ladder Operator / 昇降演算子
\[\begin{aligned} & \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} + \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} \quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \hat{x} - \frac{i}{\sqrt{2m\hbar\omega}} \hat{p} \\ & \hat{a}|n\rangle = \sqrt{n}|n-1\rangle \quad \hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle \\ & \hat{H}|n\rangle = \hbar\omega\left( n + \frac{1}{2} \right)|n\rangle \end{aligned}\]