# 連續性方程式

## 概論

### 微分形式

$\frac{\partial \varphi}{\partial t} + \nabla \cdot \mathbf{f} = s$

$\frac{\partial \varphi}{\partial t} + \nabla \cdot \mathbf{f} = 0$

### 積分形式

$\frac{\mathrm{d}Q}{\mathrm{d}t} + \oint_{\mathbb{S}} \mathbf{f}\cdot\mathrm{d}\mathbf{a} = S$

## 電磁理論

$\frac{ \partial \rho }{\partial t}+\nabla \cdot \mathbf{J} =0$

### 導引

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \ \epsilon_0 \frac{ \partial E }{\partial t}$

$0= \mu_0\nabla \cdot \mathbf{J} +\mu_0\epsilon_0 \frac{ \partial (\nabla \cdot\mathbf{E})}{\partial t}$

$\nabla \cdot\mathbf{E}=\rho/\epsilon_0$

$\frac{ \partial \rho }{\partial t} + \nabla \cdot \mathbf{J}=0$

### 四維電流

$J^{\alpha}\ \stackrel{def}{=}\ (c \rho , \mathbf{J}) = (c \rho , J_x, J_y , J_z)$

$\partial_{\alpha} J^{\alpha}=0$

## 流體力學

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$

$\nabla \cdot (\mathbf{u}) = 0$

## 能量

$\frac{ \partial u}{\partial t}+\nabla \cdot \mathbf{q}= 0$

$\mathbf{q} = -k \nabla T$

$\frac{ \partial u}{\partial t} - k\nabla^2T= 0$

## 量子力學

$\mathbf{J}\ \stackrel{def}{=}\ \frac{\hbar}{2mi}\left(\Psi^* \boldsymbol{\nabla} \Psi - \Psi \boldsymbol{\nabla} \Psi^*\right) = \frac\hbar m \mbox{Im}(\Psi^*\boldsymbol{\nabla}\Psi)$

### 連續方程式與機率保守定律

$\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla}\cdot\mathbf{J} = 0$

$\frac{\mathrm{d}}{\mathrm{d}t} \int_\mathbb{V} |\Psi|^2 \mathrm{d}^3{r} + \oint_\mathbb{S}\mathbf{J}\cdot {\mathrm{d}\mathbf{a}} = 0$(1)

### 連續方程式導引

$P= \int_\mathbb{V} \rho\,\mathrm{d}^3\mathbf{r} = \int_\mathbb{V} |\Psi|^2 \,\mathrm{d}^3\mathbf{r}$

$\frac{\mathrm{d}P}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \int_\mathbb{V} |\Psi|^2 \,\mathrm{d}^3{r} = \int_\mathbb{V} \left( \frac{\partial \Psi}{\partial t}\Psi^* + \Psi \frac{\partial \Psi^*}{\partial t} \right) \,\mathrm{d}^3{r}$(2)

$i\hbar \frac{\partial \Psi}{\partial t} = \frac{-\hbar^2}{2m} \nabla^2 \Psi + U\Psi$

$\frac{\mathrm{d}P}{\mathrm{d}t} = - \int_\mathbb{V} \frac{\hbar}{2mi} \left(\Psi^* \nabla^2 \Psi - \Psi \nabla^2 \Psi^* \right)\,\mathrm{d}^3{r}$

$\boldsymbol{\nabla} \cdot \left(\Psi^*\boldsymbol{\nabla} \Psi - \Psi \boldsymbol{\nabla} \Psi^* \right) = \boldsymbol{\nabla} \Psi^* \cdot \boldsymbol{\nabla} \Psi + \Psi^* \nabla^2 \Psi - \boldsymbol{\nabla} \Psi \cdot \boldsymbol{\nabla} \Psi^* - \Psi \nabla^2 \Psi^*$

$\frac{\mathrm{d}P}{\mathrm{d}t} = - \int_\mathbb{V} \boldsymbol{\nabla} \cdot \left[\frac{\hbar}{2mi}\left(\Psi^* \boldsymbol{\nabla} \Psi - \Psi \boldsymbol{\nabla} \Psi^* \right)\right]\,\mathrm{d}^3{r}$

$\int_\mathbb{V} \frac{\partial \rho}{\partial t}\,\mathrm{d}^3{r}= - \int_\mathbb{V} \left( \boldsymbol{\nabla} \cdot\mathbf{J}\right)\,\mathrm{d}^3{r}$

$\frac{\partial \rho}{\partial t} + \boldsymbol{\nabla} \cdot \mathbf{J} = 0$

## 參考文獻

1. ^ 1.0 1.1 1.2 Pedlosky, Joseph. Geophysical fluid dynamics. Springer. 1987: 10–13. ISBN 9780387963877.
2. ^ Clancy, L.J.(1975), Aerodynamics, Section 3.3, Pitman Publishing Limited, London