Question

The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 are simultaneous eigenfunctions of H and parity (i.e., r →-r), but they are not also simultaneous eigenfunctions of L' and Lz. However, we know that it's possible to construct eigenfunctions of H for the 3D harmonic oscillator that are also eigenfunctions of L, Lz, and parity. Combine the Gaussian factors that appear in your Prob. 4.46 eigenfunctions into a function of r that is independent of θ and φ. Then use the parities. the degeneracies d(n), the remaining powers of x, y, and z that appear in the Cartesian eigenfunctions, and the functional form of rY" to infer the allowed values of 1 and m for the states of the harmonic oscillator with energy En Note: This only requires simple arithmetic, power counting, and the functional form of Y". If you are doing a fancy calculation - or even thinking about a differential equation - yoiu are on the wrong track!

Answer 1

米→-The three-dimensional toer monic oscillator っ Suppose hat Now 2

The second term is only 요nchon The thd em is only ton m LO E.n

3 The orc d,mensional ha'moril collatov and we knoud olloud eneroie So Whext D, hert

未→ The degeneracy d(n) ct n Then tHere Then So oad