$$ \rho(\vec{r},t)=\psi^{*}(\vec{r},t)\psi(\vec{r},t) $$
$$ \frac{\partial\rho}{\partial t}=\psi^{}\frac{\partial\psi}{\partial t}+\frac{\partial\psi^{}}{\partial t}\psi $$
带入薛定谔方程即其共轭方程
$$ \frac{\partial\rho}{\partial t}=\frac{i\hbar}{2m}(\psi^\nabla^2\psi-\psi\nabla^2\psi^)=\frac{i\hbar}{2m}\nabla\cdot(\psi^\nabla\psi-\psi\nabla\psi^) $$
$$ \begin{aligned}&\nabla(\varphi\psi)=\varphi\nabla\psi+\psi\nabla\varphi \\&\nabla\cdot(\varphi \boldsymbol f)=(\nabla\varphi)\cdot \boldsymbol f+\varphi\nabla\boldsymbol{\cdot}\boldsymbol f \\&\nabla\times(\varphi\boldsymbol f)=(\nabla\varphi)\times \boldsymbol f +\varphi\nabla\times\boldsymbol f\end{aligned} $$
推导使用了二式凑出。
引入$\vec{\jmath}=\frac{i\hbar}{2m}(\psi\nabla\psi^{}-\psi^{}\nabla\psi),$化简形式为
$$ \frac{\partial\rho}{\partial t}+\nabla\cdot\vec{j}=0 $$
应用高斯定理
$$ \begin{aligned}\frac{\partial}{\partial t}\int_{\Omega}\rho d^3\vec{r}&=-\oint_{S}\vec{\jmath}\cdot d\vec{s}\end{aligned} $$
$\rho$几率密度在体积$\Omega$内的增加量等于$\vec \jmath$概率流密度矢量在界面$S$上流入量