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Homework Help: Question of spin 1/2 particles in a rotating field

  1. Jul 4, 2015 #1
    1. The problem statement, all variables and given/known data

    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]

    2. Relevant 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]
    3. 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!
    Last edited: Jul 4, 2015
  2. jcsd
  3. Jul 9, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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