Principles of Quantum Mechanics · Problem 5.3.3

The Classical Limit — R. Shankar

Physics — Quantum Mechanics · probability-current-plane-wave

Problem

Consider

\psi_p = \left(\frac{1}{2\pi\hbar}\right)^{3/2} e^{i(\mathbf{p}\cdot\mathbf{r})/\hbar}

Find

\mathbf{j}

and

P

and compare the relation between them to the electromagnetic equation

\mathbf{j} = \rho \mathbf{v}

\mathbf{v}

being the velocity. Since

\rho

and

\mathbf{j}

are constant, note that the continuity Eq. (5.3.7) is trivially satisfied.