Solution
= Solution
The <probability density> and <probability current> are
$$
\rho=|\psi|^2,
\qquad
J=\frac{\hbar}{2mi}
\left(\psi^*\frac{\partial\psi}{\partial x}
-\psi\frac{\partial\psi^*}{\partial x}\right).
$$
For a <stationary state>, $\psi(x,t)=\chi(x)e^{-iEt/\hbar}$, so $\rho=|\chi|^2$ is independent of time. The <continuity equation> then gives $\partial J/\partial x=0$. Hence
$$
\boxed{J\text{ is independent of }x}.
$$