Question of spin 1/2 particles in a rotating field

[DeathbyGreen]

Homework Statement

So I'm given a spin 1/2 particle in a rotating magnetic field in the (x,y) direction and a constant field, [itex]B_0[/itex], in the z direction and am asked to find the S matrix describing it. Given is:
[itex]B(t) = [B_1 \cos(\omega t), B_1 \sin(\omega t), B_0][/itex]

Homework Equations

[itex]H = \sum \sigma_i B_i[/itex]
[itex]H_I = H_0 + V_I[/itex]
[itex]i \hbar \frac{d}{dt} |\psi_I(t)> = H_I(t)|\psi_I(t)>[/itex]

The Attempt at a Solution

So my first thought is to plug what I know about H (Pauli matrices) and [itex]|\psi(t)>[/itex], namely that it can be decomposed into spin up and spin down states, into the interaction picture equation of motion as:

[itex]i \frac{d \psi}{dt} = V_0(t) \psi_0(t)[/itex], with [itex] V_0(t) = \exp(i H_0 t) V \exp(-iH_0 t)[/itex]

The problem is that when I plug all of this in I get an absurd looking equation that seems impossible to solve analytically. Another idea I had was to use directly

[itex] S(\infty,-\infty) = T \exp(-\int dt' V_0(t'))[/itex]

But again I get a series of strange looking exponentials that seem impossible to integrate analytically. I feel like I am making this too difficult, maybe I could just use a matrix equation like

[itex] \psi_0 (t_f) = S(t) \psi_0(t_i)[/itex]

Should I solve it perturbatively with [itex] \psi_0 (t) = \psi + \psi_1(t) + \psi_2(t)...[/itex]? Any help is appreciated!

 

[mark726]

Thank you for your post. It seems like you are on the right track with your approach using the interaction picture equation of motion. However, it is important to remember that in this case, the Hamiltonian H_0 is time-dependent due to the rotating magnetic field. Therefore, the time evolution operator in the interaction picture will also be time-dependent, which can make the equations more complicated.

One approach you could try is to use the Dyson series expansion for the time evolution operator, which can be written as:

S(t, t_0) = T \exp \left( -i \int_{t_0}^t dt' V_I(t') \right) = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 \cdots \int_{t_0}^{t_{n-1}} dt_n T \left[ V_I(t_1) V_I(t_2) \cdots V_I(t_n) \right]

where T denotes the time ordering operator. This series can be truncated at a certain order to obtain a perturbative solution to the problem. You can also try using the Magnus expansion, which is a more efficient way to calculate the time evolution operator in the case of a time-dependent Hamiltonian.

I hope this helps. Good luck with your calculations!