= Solution
The <Born rule> and <probability current> in one dimension give
$$
\rho=|\Psi|^2,
\qquad
j=\frac{\hbar}{2mi}\left(\Psi^*\Psi_x-\Psi\Psi_x^*\right).
$$
Multiply the <Time-dependent Schrödinger equation> by $\Psi^*$, subtract its complex conjugate multiplied by $\Psi$, and rearrange to obtain the <continuity equation> $\rho_t+j_x=0$.
For the stated superposition, the interference terms in $j$ are real and cancel from its imaginary part. Hence
$$
\boxed{j=\frac{\hbar k}{m}\left(|A|^2-|B|^2\right),}
$$
independent of $x$ and $t$.
Back to article page
