# 機率流

## 定義

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

### 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)

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

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

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

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

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

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

## 範例

### 平面波

$\Psi(\mathbf{r},\,t)=A e^{i\mathbf{k}\cdot\mathbf{r}} e^{i \omega t}$

$\Psi$ 的機率流是

$\mathbf{J}= \frac{\hbar}{2mi} |A|^2 \left( e^{-i\mathbf{k}\cdot\mathbf{r}} \boldsymbol{\nabla} e^{i\mathbf{k}\cdot\mathbf{r}} - e^{i\mathbf{k}\cdot\mathbf{r}} \boldsymbol{\nabla} e^{-i\mathbf{k}\cdot\mathbf{r}}\right) = |A|^2 \frac{\hbar\mathbf{k}}{m}$

### 盒中粒子

$\Psi_n = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi}{L} x \right),\qquad 0 \le x \le L$

$J_n = \frac{\hbar}{2mi}\left( \Psi_n^* \frac{\partial \Psi_n}{\partial x} - \Psi_n \frac{\partial \Psi_n^*}{\partial x} \right) = 0$