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

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1. Jul 4, 2015

### DeathbyGreen

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, $B_0$, in the z direction and am asked to find the S matrix describing it. Given is:
$B(t) = [B_1 \cos(\omega t), B_1 \sin(\omega t), B_0]$

2. Relevant equations

$H = \sum \sigma_i B_i$
$H_I = H_0 + V_I$
$i \hbar \frac{d}{dt} |\psi_I(t)> = H_I(t)|\psi_I(t)>$
3. The attempt at a solution

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

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

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

$S(\infty,-\infty) = T \exp(-\int dt' V_0(t'))$

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

$\psi_0 (t_f) = S(t) \psi_0(t_i)$

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

Last edited: Jul 4, 2015
2. Jul 9, 2015