Time dependent perturbation theory applied to energy levels

  • A
  • Thread starter BillKet
  • Start date
  • #1
BillKet
285
24
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!
 

Answers and Replies

  • #2
BillKet
285
24
@Twigg do you have any idea what they are doing here?
 
  • #3
DrClaude
Mentor
8,024
4,753
I am wondering if this analogous to calculating the AC Stark shift.
 
  • #4
BillKet
285
24
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).
 
  • #5
Twigg
Science Advisor
Gold Member
873
469
Sorry for the slow reply. I had started working it out, and I spilled tea on my papers o:)

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.
 
  • #6
BillKet
285
24
Sorry for the slow reply. I had started working it out, and I spilled tea on my papers o:)

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?
 
Last edited:

Suggested for: Time dependent perturbation theory applied to energy levels

Replies
0
Views
467
Replies
1
Views
103
Replies
13
Views
433
Replies
5
Views
283
Replies
29
Views
1K
Replies
12
Views
1K
Replies
5
Views
1K
Replies
13
Views
2K
Top