Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.15

Particle in a Gravitational Field

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

Problem

Consider the virial theorem for a particle in a power-law potential

V(z) = \lambda z^n

(for

z > 0

, with

\psi(0) = 0

). (a) State the virial theorem and use it to find a relation between

\langle T \rangle

and

\langle V \rangle

for this potential. (b) Apply the result to the gravitational case

n = 1

to find

\langle T \rangle

and

\langle V \rangle

in terms of the total energy

E

. (c) Hence calculate

\langle z \rangle

for the

n

th gravitational energy eigenstate.