# 連續性方程式

（重定向自連續方程式

## 概論

### 微分形式

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

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

### 積分形式

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

## 電磁理論

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

### 馬克士威-安培方程式滿足局域電荷守恆的連續性方程式

${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\ \epsilon _{0}{\frac {\partial E}{\partial t}}}$

${\displaystyle 0=\mu _{0}\nabla \cdot \mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial (\nabla \cdot \mathbf {E} )}{\partial t}}}$

${\displaystyle \nabla \cdot \mathbf {E} =\rho /\epsilon _{0}}$

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

### 四維電流

${\displaystyle J^{\alpha }\ {\stackrel {def}{=}}\ (c\rho ,\mathbf {J} )=(c\rho ,J_{x},J_{y},J_{z})}$

${\displaystyle \partial _{\alpha }J^{\alpha }=0}$

## 流體力學

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

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

## 能量

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

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

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

## 量子力學

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

### 連續方程式與機率守恒定律

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

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

### 連續方程式推导

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

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

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

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

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

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

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

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

## 參考文獻

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