Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.9

Particle in a Gravitational Field

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

Problem

Consider a particle of mass

m

confined between two horizontal surfaces: a floor at

z = 0

and a ceiling at

z = L

. The potential is

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

(a) Write down the boundary conditions that the wavefunction must satisfy. (b) Show that the energy eigenvalues are determined by the condition

\text{Ai}(-\zeta_0)\, \text{Bi}(-\zeta_L) - \text{Bi}(-\zeta_0)\, \text{Ai}(-\zeta_L) = 0

where

\zeta_0

and

\zeta_L

are appropriate dimensionless quantities that you should define. (c) In the limit

L \to \infty

, show that you recover the result for the semi-infinite case.