The goal of this problem is to determine degenerate eigenstates of the threedimensional isotropic harmonic oscillator written as eigenstates of L² and L_z, in terms of the Cartesian eigenstates |n_xn_yn_z) .

(a) Show that the angular-momentum operators are given by

where summation is implied over repeated indices, ε_ijk is the totally antisymmetric symbol, and N = a_j^†a_j counts the total number of quanta. 

(b) Use these relations to express the states |qlm〉 = |01m〉, m = 0, ± 1, in terms of the three eigenstates |n_xn_yn_z〉 that are degenerate in energy. Write down the representation of your answer in coordinate space, and check that the angular and radial dependences are correct.

(c) Repeat for |qlm〉 = |200〉.
(d) Repeat for |qlm〉 = |02m〉, with m = 0, 1 , and 2.

Transcribed Image Text:

Li = iħeijka jak L² =ħ² [N(N+1)—at aħaja;],