Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.1

Particle in a Gravitational Field

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

Problem

A particle of mass mm is placed in a uniform gravitational field. The Hamiltonian is H=p22m+mgzH = \frac{p^2}{2m} + mgz where gg is the acceleration due to gravity and zz is the height above the ground (which is at z=0z=0). The boundary condition is ψ(0)=0\psi(0) = 0. (a) Use dimensional analysis to show that the energy eigenvalues must have the form En=mg(ℏ22m2g)1/3anE_n = mg \left(\frac{\hbar^2}{2m^2 g}\right)^{1/3} a_n where ana_n are dimensionless numbers. (b) The Schrödinger equation in this case reduces to the Airy equation. Write down the Schrödinger equation and make the substitution z′=z−E/(mg)z' = z - E/(mg) to bring it to the standard form of the Airy equation.

Related theory

Browse the problem library