Here are some notes related to a solution of the IP2 problem, which involved exploring and elucidating the degeneracy manifolds of the 2D harmonic oscillator by finding Lz eigenstates from states constructing as products of 1D HO energy eigenstates.
What are we doing here? What is this problem all about? We can step back and take a broader perspective on this whole endeavor. This is not simple, but it may be helpful. Our central conundrum is associated with the realization that whenever there is degeneracy there are an infinite number of ways to span the degeneracy manifold. If we only look at one of them, then we have no perspective on this essential aspect, which is critical to understanding QM in more than 1D.
The things we do, like getting x,y,z states from the Psi_n,l,m and, on the other hand, getting Lz eigenstates (m) from x,y states of the 2D HO, are all related to gaining perspective in this way. Without this our perspective would be very limited.
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