Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.2

Particle in a Gravitational Field

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

Problem

Consider a particle bouncing vertically in a gravitational field, with the floor at

z = 0

acting as an infinite potential wall. The energy eigenvalues are given by

E_n = mg z_n

where

z_n

are the zeros of the Airy function

\text{Ai}(-z_n) = 0

. (a) Find the ground state energy

E_1

for a neutron (

m = 1.675 \times 10^{-27}

kg) in the Earth's gravitational field (

g = 9.8

m/s

^2

). (b) What is the classical turning point

z_1 = E_1/(mg)

for this ground state? Compare this to the size of an atom.