George Jones put up a neat little puzzle on physicsforums. Following up on this taught me quite a bit about some of the mathematical subtleties of QM that physicists tend to gloss over.
“Suppose A is an observable, i.e., a self-adjoint operator, with real eigenvalue a and normalized eigenket . In other words,
Suppose further that A and B are canonically conjugate observables, so
where I is the identity operator. Compute, with respect to , the matrix elements of this equation divided by :
In the first term, let A act on the bra; in the second, let A act on the ket:
0 = 1.
This is my favourite “proof” of the well-known equation 0 = 1.
When you’re done racking your brains, you can take a look here, here and on the arxiv for solutions to this and more such QM puzzlers. The last link is to a paper by F. Gieres called ‘Mathematical surprises and Dirac’s formalism in Quantum Mechanics’, which I found pretty fascinating. Apparently, when we’re taught as undergrads that state vectors in QM live in a Hilbert space, that’s one of those white lies that teachers tell.