Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.10

Particle in a Gravitational Field

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

Problem

A particle of mass mm is subject to a gravitational field gg and is also confined by a harmonic potential in the horizontal direction. The full 3D Hamiltonian is H=px2+py2+pz22m+12mω2(x2+y2)+mgzH = \frac{p_x^2 + p_y^2 + p_z^2}{2m} + \frac{1}{2}m\omega^2(x^2 + y^2) + mgz with the boundary condition ψ=0\psi = 0 at z=0z = 0. (a) Show that the variables separate and write the total energy as a sum of contributions from each direction. (b) Write down the allowed energies. What are the quantum numbers needed to label each state?

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