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I'm studying approximation methods, and something is really bothering me about the standard treatment of time-dependant perturbation theory.

In lecture, the prof introduced time-dependant perturbation theory with the following motivation: suppose we have a system with an initially time-independent Hamiltonian, initially in some eigenstate |i> of said Hamiltonian. At some time time t, a perturbation is turned on, which may be static or time-dependant itself, e.g. some fixed jump in the potential at time t, or a classical EM field applied at time t0. The question we ask is: what is the probability of finding the system at time t in another eigenstate |f>of the initial, unperturbed time-independant Hamiltonian? In other words, what is the probability of the system undergoing a transition from i to f?

He then goes on the develop the standard iterative method of determining said probability, involving the mag-squared inner product of |f> with the new, time-dependent state psi(t):

My problem is with the very question being asked. It seems to me ill-posed. Once the Hamiltonian is changed by the perturbation, unless the perturbation is again zero after some time, then "the probability of finding the system in state |f>", where |f> is an eigenstate of the original Hamiltonian, is (in general) zero.

My reasoning is: once the Hamiltonian is perturbed, its eigenstates change with time, and the state |f> is generally no longer an eigenstate. So a measurement of energy at time t would not collapse the system into state |f>, but rather some eigenstate of thenewHamiltonian at time t. Further, even if by chance |f> is an eigenstate at some time t, the measured energy eigenvalue could be different, if for example the perturbation at that time t was a nonzero constant.

I have consulted three standard texts: Shankar, Sakurai and Desai, but each of them states the question in essentially the same way. Finally, after turning to a great (but for some reason non-standard) book by Ballentine, I find some discussion of this:

(Leslie E. Ballentine, When problems of this sort are discussed formally, it is common to speak

of the perturbation as causing transitions between the eigenstates H0. If

this means only that the system has absorbed from the perturbing field (or

emitted to it) the energy difference ωfi = εf − εi, and so has changed

its energy, there is no harm in such language. But if the statement is

interpreted to mean that the state has changed from its initial value of

|Ψ(0) = |i to a final value of |Ψ(T ) = |f, then it is incorrect. The perturbation leads to a final state |Ψ(t), for t ≥ T , that is of the form

(12.46) with an(t) replaced by an(T ). It is not a stationary state, but

rather it is a coherent superposition of eigenstates of H0. The interference

between the terms in (12.46) is detectable, though of course it has no effect

on the probability |af (T )|2 for the final energy to be E = εf. The spinflip

neutron interference experiments of Badurek et al . (1983), which were

discussed in Sec. 12.4, provide a very clear demonstration that the effect of

a time-dependent perturbation is to produce a nonstationary state, rather

than to cause a jump from one stationary state to another. The ambiguity

of the informal language lies in its confusion between the two statements,

“the energy is εf ” and “the state is |f”. If the state vector |Ψ is of the

form (12.46) it is correct to say that the probability of the energy being

εf is |af |2. In the formal notation this becomes Prob(E = εf |Ψ) = |af |2,

which is a correct formula of quantum theory. But it is nonsense to speak

of the probability of the state being |f when in fact the state is |Ψ.Quantum Mechanics: A Modern Development(1998), pp. 351-352

This basically confirms my suspicions. It's neat to find out that the distinction has been made experimentally. However, I still have a problem: Ballentine addresses the case where the perturbation is zero outside a closed interval in time. In other texts, and in my course, we are also looking at the case where the perturbation remains in place for all time after t0. Certainly the system may then evolve to have a nonzero component along the |f> subspace, but the fact remains that a measurement of the energy will generally collapse the system to |f>, nor will it return an energy eigenvalue the same as |f>'s, in the context of H0.

Sorry if that was long or hard to follow. Let me know if I need to clarify.

Any thoughts?

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# Time-dependant perturbation theory & transitions

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