Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.18

Particle in a Gravitational Field

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

Problem

Use the variational method with trial wavefunction

\psi_\alpha(z) = \begin{cases} N z(L-z) e^{-\alpha z} & 0 \leq z \leq L \\ 0 & \text{otherwise} \end{cases}

to estimate the ground state energy of a particle of mass

m

in the potential

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

where

L

is a fixed length. Minimize the expectation value of

H

with respect to

\alpha

and compare with the exact result in the limit

L \to \infty