Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.7

Particle in a Gravitational Field

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

Problem

The Schrödinger equation for a particle in a linear potential

V = Fz

(where

F

is a constant force) can be transformed into the Airy equation. (a) Make the substitution

u = \left(\frac{2mF}{\hbar^2}\right)^{1/3}\left(z - \frac{E}{F}\right)

and show that the Schrödinger equation becomes

\frac{d^2\psi}{du^2} = u\,\psi

(b) The general solution is

\psi = c_1 \text{Ai}(u) + c_2 \text{Bi}(u)

. For a particle confined to

z > 0

with

V = \infty

z = 0

, explain why we must set

c_2 = 0

for the solution that is physically acceptable as

z \to \infty