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In summary: If you started in the unperturbed state and calculated the expectation value of the Hamiltonian, then that would be equivalent to calculating the energy in that state. The state you are in is not necessarily an eigenstate of the new Hamiltonian, but it would give you the energy in that state.

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

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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).DrClaude said:I am wondering if this analogous to calculating the AC Stark shift.

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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.

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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?).Twigg said:

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.

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?

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