- #1

MichPod

Gold Member

- 189

- 32

- Summary:
- Hard to express, but smth like "an apparent problem in choosing between a "big" Hilbert space of quantum states with non-countable basis and a "small" Hilbert space generated by a countable set of basis vectors".

It's probably more kind of math question.

I consider a wave function of a harmonic oscillator, i.e. a particle in a parabolic well of potential. We know that the Hamiltonian is a Hermitian operator, and so its eigenstates constitute a full basis in the Hilbert space of the wave function states. We also know that this basis is countable.

On the other side, the arbitrary state may also be considered as a weighted integral of delta-functions, and such delta-functions are obviously a non-countable basis which creates a "bigger" space than what the eigenstates of the original Hamiltonian could generate.

I then wonder what should be called a "true" Hilbert space for such a case? A continuous spectrum of eigenfunctions is common in QM, it is generated for instance by a "free" (no potential part) Hamiltonian, so it's not a completely weird idea that the space of the function should have uncountable basis for practically ANY potential, including parabolic - after all, the wave function at a single moment may be arbitrary. Also a question may be asked, how a delta-wave-function would evolve in a parabolic well potential, but such a function obviously lies out of the subspace generated by the countable set of eigenstates of the Hamiltonian of the parabolic potential, so how such a problem should be approached, while we are used to a countable basis for the Harmonic oscillator?

I consider a wave function of a harmonic oscillator, i.e. a particle in a parabolic well of potential. We know that the Hamiltonian is a Hermitian operator, and so its eigenstates constitute a full basis in the Hilbert space of the wave function states. We also know that this basis is countable.

On the other side, the arbitrary state may also be considered as a weighted integral of delta-functions, and such delta-functions are obviously a non-countable basis which creates a "bigger" space than what the eigenstates of the original Hamiltonian could generate.

I then wonder what should be called a "true" Hilbert space for such a case? A continuous spectrum of eigenfunctions is common in QM, it is generated for instance by a "free" (no potential part) Hamiltonian, so it's not a completely weird idea that the space of the function should have uncountable basis for practically ANY potential, including parabolic - after all, the wave function at a single moment may be arbitrary. Also a question may be asked, how a delta-wave-function would evolve in a parabolic well potential, but such a function obviously lies out of the subspace generated by the countable set of eigenstates of the Hamiltonian of the parabolic potential, so how such a problem should be approached, while we are used to a countable basis for the Harmonic oscillator?