For the three-dimensional isotropic harmonic oscillator

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

For the three-dimensional isotropic harmonic oscillator H^=∑i=x,y,z(p^i2/2m+12mω2x^i2)\hat H=\sum_{i=x,y,z}\big(\hat p_i^2/2m+\tfrac12m\omega^2\hat x_i^2\big), consider the excited level with total quantum number N=3N=3 (energy E3=(3+32)ℏωE_3=(3+\tfrac32)\hbar\omega). ((a)) Using the Cartesian counting of oscillator states, compute the total degeneracy g3g_3 of this level. ((b)) This level decomposes into a direct sum of angular-momentum multiplets with l=N,N−2,…l=N,N-2,\dots down to l=1l=1 or l=0l=0. List the values of ll present for N=3N=3 together with their multiplet dimensions 2l+12l+1, and verify that these dimensions sum to g3g_3. ((c)) Rotational symmetry alone, [H^,L⃗^]=0[\hat H,\hat{\vec L}]=0, guarantees only that each individual ll-multiplet found in part (b) is (2l+1)(2l+1)-fold degenerate. Explain why it says nothing about why the different ll multiplets found should share the same energy, and name the larger symmetry group, and the operators generating it, that is responsible for this extra degeneracy.

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