Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.3

Particle in a Gravitational Field

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

Problem

The WKB approximation gives the quantization condition for a particle bouncing in a gravitational field:

\int_0^{z_n} \sqrt{2m(E_n - mgz)}\, dz = \left(n - \frac{3}{4}\right)\pi\hbar

(a) Evaluate the integral on the left-hand side. (b) Show that the WKB energy eigenvalues are

E_n^{\text{WKB}} = mg\left[\frac{3\pi}{2}\left(n - \frac{3}{4}\right)\right]^{2/3} \left(\frac{\hbar^2}{2m^2 g}\right)^{1/3}

n = 1, 2, 3