Time dependent perturbation theory applied to energy levels

[BillKet]

Hello! I am reading this paper and in deriving equations 6/7 and 11/12 they claim to use second oder time dependent perturbation theory (TDPT) in order to get the correction to the energy levels. Can someone point me towards some reading about that? In the QM textbooks I used, for TDPT they just calculate the change in population as a function of time, but I have never seen a formula for the change in energy levels. I am able to derive 6/7 and 11/12 by applying a unitary transformation to the hamiltonian and working from there, but is there a simple formula to get these equations directly (similar to the energy formula for time independent perturbation theory)? Thank you!

 

[BillKet]

@Twigg do you have any idea what they are doing here?

 

[DrClaude]

I am wondering if this analogous to calculating the AC Stark shift.

 

[BillKet]

I am wondering if this analogous to calculating the AC Stark shift.

From what they claim in the paper, that seems to be the case, but I still don't know how to derive this perturbation theory formula in general (for a 2x2 level system at least).

 

[Twigg]

Sorry for the slow reply. I had started working it out, and I spilled tea on my papers

I don't think there is a direct formula. The way I was working it out was getting the spin-up and spin-down populations as a function of time from 2nd order TDPT, then taking the expectation value over the Hamiltonian.

If you wrote down the expectation value and substituted in the TDPT formulae for the perturbative corrections to the wavefunction, you would end up with a direct formula but it would be lengthy to the point of uselessness.

 

[BillKet]

Sorry for the slow reply. I had started working it out, and I spilled tea on my papers

I don't think there is a direct formula. The way I was working it out was getting the spin-up and spin-down populations as a function of time from 2nd order TDPT, then taking the expectation value over the Hamiltonian.

If you wrote down the expectation value and substituted in the TDPT formulae for the perturbative corrections to the wavefunction, you would end up with a direct formula but it would be lengthy to the point of uselessness.

That looks very tedious (unless I am doing something wrong), and it also requires doing several integrals (in this case they are easy but in general it can be very difficult, no?).

Also I am a bit confused about this. If I start in an eigenstate of the unperturbed Hamiltonian, say $(1,0)$, after a time, t, to second order in PT I will be in a state $c_a(t)(1,0)+c_b(t)(0,1)$. Now I would calculate the expectation value of the Hamiltonian in this state and get the energy. But is this state $c_a(t)(1,0)+c_b(t)(0,1)$ an eigenstate of the new Hamiltonian such that the expectation value can be interpreted as an energy? Shouldn't I diagonalize my time dependent Hamiltonian, get the eigenvectors, and then propagate them in time? Or are the 2 approaches equivalent?