A Time dependent perturbation theory applied to energy levels

BillKet
Messages
311
Reaction score
30
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!
 
Physics news on Phys.org
@Twigg do you have any idea what they are doing here?
 
I am wondering if this analogous to calculating the AC Stark shift.
 
DrClaude said:
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).
 
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.
 
Twigg said:
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:
Hi. I have got question as in title. How can idea of instantaneous dipole moment for atoms like, for example hydrogen be consistent with idea of orbitals? At my level of knowledge London dispersion forces are derived taking into account Bohr model of atom. But we know today that this model is not correct. If it would be correct I understand that at each time electron is at some point at radius at some angle and there is dipole moment at this time from nucleus to electron at orbit. But how...

Similar threads

Replies
0
Views
1K
Replies
156
Views
10K
Replies
13
Views
5K
Replies
0
Views
3K
Replies
3
Views
1K
Replies
1
Views
2K
Replies
1
Views
2K
Back
Top