I've been working my way through some basic quantum mechanics, and have gotten up to perturbation theory. It basically makes sense to me, but there's one thing that bothers me, and I was wondering if somebody could shed some light on it.(adsbygoogle = window.adsbygoogle || []).push({});

The essential idea behind perturbation theory is that we start with a basic Hamiltonian [tex]\hat{H}^0[/tex], and solve for its energy levels:

[tex]\hat{H}^0_n\Psi^0_n = E^0_n\Psi^0_n[/tex]

We then form a new Hamiltonian by applying a perturbation, to form [tex]\hat{H} = \hat{H}^0 + \lambda\hat{H}'}[/tex], and attempt to solve for its energy levels.

It looks as though the technique attempts to do this by assuming that the eigenfunctions of the full Hamiltonian, [tex]\Psi[/tex] can be expressed as a linear combination of the set of base eigenfunctions, [tex]\Psi^0_n[/tex], and it's here that I have a question. That assumption is true if the eigenfunctions of [tex]\hat{H}^0[/tex] form a complete basis, but is that always the case? If [tex]\hat{H}^0[/tex] is something like a potential well, then it will only have a discrete set of eigenvalues, not a continuous range. Is it still possible for a linear combination of these discrete eigenfunctions to completely span the state space?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Energy eigenfunctions in time-independent perturbation theory

**Physics Forums | Science Articles, Homework Help, Discussion**