# What happens with the momentum properties of the finite-well eigenstates as the well becomes deeper and deeper?

It's a reasonably standard fact (though not nearly well-known enough) that the momentum operator in the infinite square well is a very problematic beast (as explained e.g. in this and this answers).

In the comments under this answer, mike stone proposed an interesting look at how this might work:

[the] question can be formulated more physically by asking what happens in finite depth square well on the entire line. Then a self-adjoint momentum operator exists (but does not commute with the Hamiltonian) and the overlaps do give the possible momenta, [and] one can then explore what happens as the depth increases.

I think that's an excellent way to look at it, so: what are the momentum properties of the finite-well eigenstates, and how do they change in the limit of a very tall well? How do they mesh with the properties of the (possible self-adjoint extensions of the) momentum operator on the infinite square well, and its interactions with the hamiltonian over the restricted-interval version?

• As a side note, one might be also interested in the related the problem of self-adjointness of the Hamiltonian in an infinite well as discussed in arxiv.org/pdf/quant-ph/0103153.pdf, where it is shown that $H=p^2/2m$ is not self-adjoint in this case. Also related is an extension of the above discussion available from fma.if.usp.br/~piza/artigos/001401ajp_Chico.pdf – ZeroTheHero Oct 10 '17 at 17:54
• @ZeroTheHero Yeah, I looked for good references for that earlier today and didn't find any as easily as I'd've liked. We're probably overdue a good canonical question, with solid and comprehensive references, on the status of the momentum operator in the infinite square well. – Emilio Pisanty Oct 10 '17 at 18:03
• There's a lot of related literature that cites Bonneau so Gsearching for this paper in GScholar is a good first step. I think the explicit case of the momentum operator is discussed in Chap.6 of the QM book by Capri. – ZeroTheHero Oct 10 '17 at 18:43
• @ZeroTheHero Yeah, the information definitely exists - I'm just saying that this site ought to have a concise, clear and complete summary. – Emilio Pisanty Oct 10 '17 at 20:19
• Amen to that!!! – ZeroTheHero Oct 10 '17 at 20:28