# A Oscillator Model with Eigenfunctions

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1. Feb 11, 2019 at 11:43 AM

### Vajhe

Hi, I have been reading the Milonni and Eberly book "Laser": in one of the chapters they discuss the Oscillator Model. The treatment is quite straightforward, the Hamiltonian of the process is

H=H0+HI

where the first term is the "undisturbed" hamiltonian, and the second one is the interaction produced by an applied field.

Φn are the eigenfunctions that solve the Schrödinger equation with HI=0 (i.e no applied field), but to continue you have to assume that the same functions (Φn) are also solutions to the problem when there is an applied field. How can you say that? What do you lose? Is there some kind of rule of thumb to be able to say that two problems (not necessarily these) will share eigenfunctions?

2. Feb 11, 2019 at 7:32 PM

### king vitamin

You've almost certainly seen this in an introductory quantum mechanics course: if two operators share the same set of eigenfunctions, they necessarily commute. So for your case, $[H_0,H_I] = 0$. Be sure you know how to prove this statement!

3. Feb 14, 2019 at 12:44 PM

### Vajhe

Oh, I was looking the answer in the wrong place: I will have to remove the dust from my Sakurai. Thanks a lot!