- #1

andrewryno

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## Homework Statement

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

[tex]H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}[/tex]

Find the probability that the particles' spin flips at [itex]t \rightarrow \infty[/itex].

## Homework Equations

[tex]H_0 = \omega_0 S_z[/tex]

[tex]H_1 = \omega_1 S_x e^{\frac{-t}{\tau}}[/tex]

Transition probability:

[tex]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[/tex]

## 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.