= Solution
The <Time-dependent Schrödinger equation>, with the <Hamiltonian> for a particle of <mass> $m$ in a real <potential energy> $V$, is
$$
i\hbar\frac{\partial\Psi}{\partial t}=H\Psi,\qquad H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x).
$$
A <stationary state> is an <energy eigenstate>: its <wavefunction> evolves by a single <global phase>,
$$
\Psi(x,t)=\psi(x)e^{-iEt/\hbar},\qquad \boxed{-\frac{\hbar^2}{2m}\psi''+V\psi=E\psi.}
$$
Its <probability density> is independent of time. For a normalized <wavefunction>, the <probability density> and <probability current> are
$$
\boxed{\rho=\Psi^*\Psi,\qquad j=\frac{\hbar}{2mi}(\Psi^*\partial_x\Psi-\Psi\partial_x\Psi^*)=\frac{\hbar}{m}\operatorname{Im}(\Psi^*\partial_x\Psi).}
$$
Subtracting the <complex conjugate> of the <Time-dependent Schrödinger equation>, multiplied by $\Psi$, from the original equation multiplied by $\Psi^*$ cancels the real <potential energy> terms and gives the <continuity equation> $\partial_t\rho+\partial_xj=0$. Thus $j$ measures signed probability flow in the positive $x$ direction.
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