Principles of Quantum Mechanics · Problem 10.2.3

Systems with N Degrees of Freedom — R. Shankar

Physics — Quantum Mechanics · three-dimensional-isotropic-oscillator

Problem

Quantize the three-dimensional isotropic oscillator for which

\mathscr{H}=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{1}{2}m\omega^2(x^2+y^2+z^2)

(1) Show that

E=(n+3/2)\hbar\omega

;

n=n_x+n_y+n_z

;

n_x,n_y,n_z=0,1,2,\ldots

(2) Write the corresponding eigenfunctions in terms of single-oscillator wave functions and verify that the parity of the level with a given

n

(-1)^n

. Reexpress the first four states in terms of spherical coordinates. Show that the degeneracy of a level with energy

E=(n+3/2)\hbar\omega

(n+1)(n+2)/2