- #1
andrewryno
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I'm rather stuck on this problem. I seem to be having issues with the simplest things on this when trying to get started.
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].
[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]
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.
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.