贡献者: addis
一维情况下,对于某个波函数 $\psi(x,t)$,定义概率流为
\begin{equation}
j(x,t) = \frac{ \mathrm{i} \hbar}{2m} \left(\psi \frac{\partial \psi^*}{\partial x} - \psi^* \frac{\partial \psi}{\partial x} \right)
= \frac{\hbar}{m} \operatorname{Im} \left[\psi^* \frac{\partial}{\partial{x}} \psi \right] ~.
\end{equation}
某个区间中的概率增加率等于流入该区间的概率流
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{t}} P_{ab}(t) = j(a,t) - j(b,t)~.
\end{equation}
三维情况下,概率流的定义变为
\begin{equation}
\boldsymbol{\mathbf{j}} ( \boldsymbol{\mathbf{r}} ,t) = \frac{ \mathrm{i} \hbar }{2m} (\psi \boldsymbol\nabla \psi^* - \psi ^* \boldsymbol\nabla \psi)
= \frac{\hbar}{m} \operatorname{Im} [\psi^* \boldsymbol\nabla \psi]~,
\end{equation}
且有
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{t}} P_\mathcal{V}(t) = \frac{\mathrm{d}}{\mathrm{d}{t}} \int_\mathcal{V} \left\lvert \psi ( \boldsymbol{\mathbf{r}} ,t) \right\rvert ^2 \,\mathrm{d}{V}
= \int_\mathcal{S} \boldsymbol{\mathbf{j}} ( \boldsymbol{\mathbf{r}} ,t) \boldsymbol\cdot \,\mathrm{d}{ \boldsymbol{\mathbf{s}} } ~,
\end{equation}
或写为概率守恒公式(类比
电荷守恒)
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{t}} (\psi^* \psi) + \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathbf{j}} = \boldsymbol{\mathbf{0}} ~.
\end{equation}
习题 1 平面波
求三维平面波 $A \exp\left( \mathrm{i} \boldsymbol{\mathbf{k}} \boldsymbol\cdot \boldsymbol{\mathbf{r}} \right) $ 的概率流密度。答案:$ \left\lvert A \right\rvert ^2 \hbar \boldsymbol{\mathbf{k}} /m$,注意这恰好等于概率密度乘以粒子速度(原因见下文)。
习题 2 球面波
求球面波 $A \exp\left( \mathrm{i} k r\right) /r$ 的概率流密度($r = \left\lvert \boldsymbol{\mathbf{r}} \right\rvert $)。答案:$ \left\lvert A \right\rvert ^2 \hbar \boldsymbol{\mathbf{k}} /(mr^2)$,同样等于概率密度乘以粒子速度,注意单位时间通过任意球面的概率都是一样的。
1. 推导
对一维情况有
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{t}} P_{ab} = \frac{\mathrm{d}}{\mathrm{d}{t}} \int_a^b \psi^* \psi \,\mathrm{d}{x} = \int_a^b \left(\psi \frac{\partial}{\partial{t}} \psi^* + \psi^* \frac{\partial}{\partial{t}} \psi \right) \,\mathrm{d}{x} ~.
\end{equation}
一维薛定谔方程以及复共轭为
\begin{equation}
\mathrm{i} \hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar ^2}{2m} \frac{\partial^{2}{\psi}}{\partial{x}^{2}} + V\psi~,
\end{equation}
\begin{equation}
- \mathrm{i} \hbar \frac{\partial \psi^*}{\partial t} = - \frac{\hbar ^2}{2m} \frac{\partial^{2}{\psi^*}}{\partial{x}^{2}} + V{\psi^*}~.
\end{equation}
代入上式的时间微分,得
\begin{equation} \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}{t}} P_{ab} &= \frac{ \mathrm{i} \hbar }{2m} \int_a^b \left(\psi^* \frac{\partial^{2}{\psi}}{\partial{x}^{2}} - \psi \frac{\partial^{2}{\psi^*}}{\partial{x}^{2}} \right) \,\mathrm{d}{x} = \frac{ \mathrm{i} \hbar }{2m} \int_a^b \frac{\partial}{\partial{x}} \left(\psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} \right) \,\mathrm{d}{x} \\
&= \left. \frac{ \mathrm{i} \hbar }{2m} \left(\psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} \right) \right\rvert _{x=a}^{x=b} = j(a) - j(b)~,
\end{aligned} \end{equation}
三维情况的证明可类比。
2. 概率的速度
类比经典力学或电磁学中的 $ \boldsymbol{\mathbf{j}} = \rho \boldsymbol{\mathbf{v}} $,若定义概率流速度为概率流除以概率密度,则平面波 $\psi (x) = A \mathrm{e} ^{ \mathrm{i} \boldsymbol{\mathbf{k}} \boldsymbol\cdot \boldsymbol{\mathbf{r}} }$ 的概率流速为
\begin{equation}
\boldsymbol{\mathbf{v}} = \boldsymbol{\mathbf{j}} / \left\lvert \psi \right\rvert ^2 = \frac{ \mathrm{i} \hbar}{2m} \left(- \left\lvert A \right\rvert ^2 \mathrm{i} \boldsymbol{\mathbf{k}} - \left\lvert A \right\rvert ^2 \mathrm{i} \boldsymbol{\mathbf{k}} \right) / \left\lvert A \right\rvert ^2 = \frac{\hbar \boldsymbol{\mathbf{k}} }{m} = \frac{ \boldsymbol{\mathbf{p}} }{m} = \boldsymbol{\mathbf{v}} _{CM}~.
\end{equation}
所以平面波的概率流速度等于具有相同动量的经典粒子的速度。