Principles of Quantum Mechanics · Problem 10.2.2

Systems with N Degrees of Freedom — R. Shankar

Physics — Quantum Mechanics · two-dimensional-oscillator

Problem

Quantize the two-dimensional oscillator for which

\mathscr{H}=\frac{p_x^2+p_y^2}{2m}+\frac{1}{2}m\omega_x^2 x^2+\frac{1}{2}m\omega_y^2 y^2

(1) Show that the allowed energies are

E=(n_x+1/2)\hbar\omega_x+(n_y+1/2)\hbar\omega_y,\quad n_x,n_y=0,1,2,\ldots

(2) Write down the corresponding wave functions in terms of single oscillator wave functions. Verify that they have definite parity (even/odd) number

x\to-x

y\to-y

and that the parity depends only on

n=n_x+n_y

. (3) Consider next the isotropic oscillator (

\omega_x=\omega_y

). Write explicit, normalized eigenfunctions of the first three states (that is, for the cases

n=0

and 1). Reexpress your results in terms of polar coordinates

\rho

and

\phi

(for later use). Show that the degeneracy of a level with

E=(n+1)\hbar\omega

(n+1)