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It is usually the case that a relevant conservation law must exist because of some underlying symmetry. For example, I can understand that the selection rule "Δn ∈ evens" is a consequence of parity conservation, which is in turn a consequence of inversion symmetry.

My question is this: Is there an underlying symmetry in the Hamiltonian forcing the selection rule "Δn = 0" to exist? What is it?

I suppose my best guess is that it has something to do with momentum conservation, although it's a little hard to wrap my head around this since the quantity conserved is not "k" in the usual sense, but rather "k_nz = n*pi/w", (where "n" is a positive integer and "w" is the well width), and there is no translation symmetry along the "z" direction. Maybe there's a symmetry related to continuous transformations of the well width?