Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.20

Particle in a Gravitational Field

Physics — Landau & Lifshitz · particle-in-gravitational-field

Problem

The probability current for a particle in the gravitational potential

V = mgz

j = \frac{\hbar}{2mi}\left(\psi^*\frac{\partial\psi}{\partial z} - \psi\frac{\partial\psi^*}{\partial z}\right)

(a) For an energy eigenstate, show that

j = 0

everywhere (as expected for a stationary state). (b) Consider a general superposition

\psi(z,t) = c_1 \psi_1(z) e^{-iE_1 t/\hbar} + c_2 \psi_2(z) e^{-iE_2 t/\hbar}

. Calculate

j(z,t)

and show that it oscillates with angular frequency

(E_2 - E_1)/\hbar