Franz Schwabl Quantum Mechanics 4th Edit · Problem 11.3

Relativistic Corrections

Physics — Quantum Mechanics · two-dimensional-oscillator-perturbation-theory

Problem

In dimensionless variables

\left(\hbar = 1, m = 1,\text{ and thus } p_j = \frac{1}{\mathrm{i}}\frac{\partial}{\partial x_j}\right)

the Hamiltonian of a two-dimensional oscillator takes the form

H = \frac{1}{2}\left(p_x^2 + p_y^2\right) + \frac{1}{2}(1 + \delta xy)(x^2 + y^2) \quad ,

where we suppose that

\delta \ll 1

. Determine the wave functions for the three lowest lying energy levels in the case

\delta = 0

. Calculate the shift of these levels for

\delta \neq 0

in first-order perturbation theory. Note the degeneracy which occurs.