Does quantum mechanics predict Universe is infinite? because otherwise every wavefunction would be normalisable

Question

I studied the hydrogen atom recently and found that reason for discrete energy level is simply because of the fact that eigenfunction corresponding to the energy between ground state and first excited state is not normalizable. But if the Universe is finite then every wave function would be normalizable and makes it possible for any energy state to exist, which will lead to contradiction. Further the wave function of ground state is non zero for arbitrary long distance which says Universe must be infinite. This is my first post and sorry if I posted something wrong.

Answer 1

No, for two reasons:

1. The actual, measured frecuencies are never actually discrete. They have a natural width.

2. A particle in a box has discrete eigenvalues as well, so discreteness is mostly independent of the finiteness of the system.

In the case of the hydrogen atom, you can also put your particle in a box. In this case, the system is finite, and all (integrable) functions are normalisable, but the energies are still discrete. In fact, the quantisation of levels is obtained by imposing a set of boundary conditions (and univaluedness, etc.).