- #1

Manchot

- 473

- 4

Last year, I took my university's undergraduate QM sequence. We mainly used Griffiths' book, but we also used a little of Shankar's. Anyway, I decided to go through Shankar's book this year, in a more formal treatment of QM. After the first chapter, I already have some questions that I hope can be answered:

1. When extending the results from finite-dimensional vector spaces to the infinite-dimensional spaces, it was assumed that most of the same results would hold. I really don't have a big conceptual problem with this, except in the case of matrix elements. Shankar shows that Hermiticity of the matrix elements of an operator in an infinite-dimensional space is not a sufficient condition for that operator to be Hermitian; namely, the surface term must also vanish for all vectors in the space. I can see the mathematical reason for this, but conceptually, what is the reason? My guess is that because the basis vectors are not well-defined at the endpoints, the matrix elements are not well-defined there as well (e.g., in position space, the basis vector at the endpoint is only "half" a delta function). Is this line of reasoning correct?

2. All throughout QM last year, I was deeply troubled by the fact that the boundary conditions were applied during the calculation of eigenvectors rather than after. For example, in the calculation of the states of the infinite square well, I didn't like that the endpoint-vanishing conditions were applied when calculating the eigenvectors (and quantizing them) as opposed to summing over all the continuous range of eigenvectors and then applying the same results (getting delta functions at the quantized values). I never really understood where this comes from, so I'd like to now venture a guess. Does this just arise from the definition of physical Hilbert space? As the surface term must vanish for the Hamiltonian to be Hermitian, one might conclude that to get a physical system, one must force the surface terms to be zero for all vectors in the space (including the eigenvectors, of course). That would explain a lot to me, but not quite everything. For example, what about the continuity of the wavefunction? Again, that could be explained away by incorporating continuity into the definition of physical Hilbert space. However, I wouldn't see the motivation behind doing this. Finally, what about the near-continuity of the derivative of the wavefunction? Since it depends on the value of the Hamiltonian at certain points, I wouldn't think that this could be logically incorporated into the definition of Hilbert space very easily.

Can anyone resolve these issues for me? If so, I would greatly appreciate it.