Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.6

Particle in a Gravitational Field

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

Problem

A quantum bouncing ball is a particle in the potential

V(z) = \begin{cases} mgz & z \geq 0 \\ \infty & z < 0 \end{cases}

(a) Show that in the semiclassical limit (large

n

), the spacing between energy levels

E_{n+1} - E_n

approaches the classical oscillation frequency

\omega_n

times

\hbar

, i.e.,

E_{n+1} - E_n \approx \hbar \omega_n

. (b) Find the classical period of oscillation for a particle with energy

E_n

and verify this correspondence.