In this question I suspect some of the used words are not precise so there is a possibility for misunderstanding here. If you know how to say more correctly or precisely - EDIT! Sorry in advance.


Consider a particle in the following 1D potential well $$ U(x) = \begin{cases} 0, & x\in[0,b]\\ V, &\text{otherwise} \end{cases} \tag{*} \label{pot} $$ (1st problem) If $V<\infty$ then there are both discrete and continuum spectra. So Hilbert space corresponding to this problem is non-countably-infinite-dimensional.

(2nd problem) But in the case of an infinite well potential there is only discrete spectrum, so Hilbert space is countably-infinite-dimensional. So these Hilbert spaces are different, not even isomorphic.


On the one hand, if one increases $V$ continuously (e.g. in time) in eq.\eqref{pot} the first problem becomes "closer", in some sence, to the second one. And we should (shouldn't we?) conclude that the second one is a limit for the first one. But, on the other hand, two problems are the same implies their Hilbert spaces are equal (maybe isomorphic). I am wondering

How is it possible that a continuum transformed into a countable set (sorry again) "continuously"?


What about the energy values? Considering the first problem there are both discrete and continuum spectra: $$ E^{(1)} = \{E_i|i\in \mathbb{N}\}\cup\{E_\alpha|\alpha\in\mathbb{R}\} $$ so $\mathrm{card}(E^{(1)})=\aleph_1$. The energy spectrum in the second problem consists only of discrete values so $\mathrm{card}(E^{(2)})=\aleph_0$. So how and where the continuum disappears when $V$ approaches to infinity?


1 Answer 1


In both problems the Hilbert space is $L^2 (\mathbb R)$ which is an infinite-dimensional complex vector space and is separable, i.e. it has a countable basis (Schauder basis). The "(non)-countably-infinite-dimensional" phrase you use is imprecise. I assume you mean that the basis might be countable or not, case in which I said the Hilbert space has a countable Schauder basis.

I sense that you vaguely hint to the concept of rigged Hilbert spaces. A readable account of them is the textbook on Quantum Mechanics by Leslie Ballentine, or a more involved one is in the textbook by Galindo and Pascual (1st volume of the English edition published by Springer Verlag).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.