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First order time dependent perturbation theory problem

A system is in an eigenstate | psi_i > with energy E_i. The perturbation

V(t) = H'exp(-((alpha)^2)(t^2))

is turned on at t_i = -infinity and left on until t_f = infinity. Here H' is independent of time, and alpha is a constant. Show that at t_f = infinity, the probability that the system has evolved into the eigenstate | psi_f > with energy E_f is

P(i -> f) = (pi)/((hbar^2)(alpha^2)|< psi_f | H' |psi_i >|^2 exp[-((E_f - E_i)^2)/(2(hbar^2)(alpha^2)].

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Solution Preview

If we denote the time dependent wave function of the system by |psi(t)> and the eigenstates with energy E_r by |r>, then we can write:

|psi> = sum over r of c_r (t) exp(-E_r t/hbar) |r>

The coefficients c_r(t) would not depend on time if there were no time dependent perturbation. If the system is initially in the state |i>, then that means that initially c_i = 1 and all other coefficients are zero. If we compute the coefficients in perturbation theory, we find ...

Solution Summary

We consider a time dependent perturbation of the form

V(t) = H'exp[-((alpha)^2)(t^2)]

and compute to first order in perturbation theory the transition probability for an eigenstate with energy E_1 at
t = -infinity to evolve to an eigenstate with energy E_2 at t = infinity.