For the isotropic three-dimensional oscillator the shell has energy

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation in three dimensions and central potentials

Problem

For the isotropic three-dimensional oscillator the shell N=nx+ny+nzN = n_x + n_y + n_z has energy (N+32)ℏω(N + \tfrac32)\hbar\omega, degeneracy 12(N+1)(N+2)\tfrac12(N+1)(N+2) and parity (−1)N(-1)^N; the state (x+iy)N(x + iy)^N times the ground-state Gaussian lies in the shell and has ml=Nm_l = N, and no larger ll than NN occurs. Take it as given that each value of ll that occurs in a shell occurs exactly once (one complete set of 2l+12l+1 states).Consider the shell containing the state (x+iy)4(x + iy)^4 times the ground-state Gaussian. ((a)) How many Cartesian states does this shell hold? ((b)) Which values of ll does the shell contain? Give them largest first, using the parity of the shell and the largest ll present. ((c)) Check your answer to (b) against the count in (a) by adding the sizes 2l+12l+1 of the sets.

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