Graduate Oscillator Model with Eigenfunctions

[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+H_I

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

Φ_n are the eigenfunctions that solve the Schrödinger equation with H_I=0 (i.e no applied field), but to continue you have to assume that the same functions (Φ_n) 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?

 

[king vitamin]

Is there some kind of rule of thumb to be able to say that two problems (not necessarily these) will share eigenfunctions?

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!

 

[Vajhe]

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