Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.16

Particle in a Gravitational Field

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

Problem

A particle of mass

m

is released from rest at height

z_0

above the ground (which is a perfectly reflecting hard floor). The initial state is a Gaussian wave packet

\psi(z, 0) = \left(\frac{1}{2\pi\sigma^2}\right)^{1/4} \exp\left(-\frac{(z-z_0)^2}{4\sigma^2}\right)

(a) In the classical limit (

\sigma \ll z_0

), describe the subsequent motion. (b) For a neutron with

z_0 = 10\,\mu

m and

\sigma = 1\,\mu

m, after how many bounces would quantum effects become important? (Use the characteristic gravitational length

z_0^{\text{quant}} = (\hbar^2/2m^2g)^{1/3}

as a measure of quantum effects.)