Time-Dependent Perturbation Theory and Transition Probabilities

[andrewryno]

I'm rather stuck on this problem. I seem to be having issues with the simplest things on this when trying to get started.

Homework Statement

There is a particle with spin-1/2 and the Hamiltonian H_0 = \omega_0 S_z. The system is perturbed by:

H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}

Find the probability that the particles' spin flips at t \rightarrow \infty.

Homework Equations

H_0 = \omega_0 S_z
H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}

Transition probability:
P_{if}\left(t\right) = \left|-\frac{i}{\hbar} \int_0^t <\psi_f\left|V\left(t'\right)\right|\psi_i> e^{i \omega_{fi} t'} dt'\right|^2

The Attempt at a Solution

Unfortunately I don't really have any concrete math to post in this section. I understand the derivation of the transition probability--looked it over both in Griffiths and Zettili (as well as my professors notes, which are mostly incomprehensible). I understand that I need to take the perturbation (H_1, or V in the probability equation), plug that in and replace the final and initial states with the ground and flipped states for the spin 1/2 particle. That's the main part I'm getting stuck at (probably one of many issues I'll face in this homework).

Now, I could probably solve the Hamiltonian for a general wavefunction in the ground state, but I feel like doing that wouldn't help me at all. I just figured that I could change the bra-ket notation in the formula to the integral notation and solve that (one of the examples in the book did it, but for an infinite well instead). I could probably then do the same thing for the state where the spin is flipped. Is that the right way to approach it? If not, I'm completely out of ideas.

 

[TSny]

I would start by working out what V(t)|\psi_i> = H_1(t)|\uparrow>_z yields.