## time evolution of spin state

1. The problem statement, all variables and given/known data

An +x-polarized electron beam is subjected to magnetic field in the y-direction. What is the probablity of measuring spin +x after a period of time t.

2. Relevant equations

Time evolution operator $U = e^{-i/\hbar \hat{H} t}$

3. The attempt at a solution

Since the magnetic field is in the y-direction, the corresponding Hamiltonian is of the form $\hat{H} = - \gamma B_0 \hat{S}_y$. The energy eigenvalues of this are just $-\gamma B_0$ times the eigenvalues for the Spin-y operator, ie $\pm \frac{\gamma B_0 \hbar}{2}$ where $|S_y ; + \rangle =\frac{1}{\sqrt{2}}(-i,1)^T, |S_y,- \rangle = \frac{1}{\sqrt{2}}(i,1)$.

$|S_x;+ \rangle = 2\left(\frac{1+i}{4}|S_y;+ \rangle + \frac{1-i}{4}|S_y;-\rangle\right)$

so the time evolved state vector is

$|S_x;+ \rangle = 2\left(\frac{1+i}{4} e^{i t\gamma B_0 \hbar/2}|S_y;+ \rangle + \frac{1-i}{4}e^{-i t\gamma B_0 \hbar/2}|S_y;-\rangle\right)$

$|S_x;+ \rangle = \cos (t \gamma B_0 \hbar/2)2\left(\frac{1+i}{4} |S_y;+ \rangle + \frac{1-i}{4}|S_y;-\rangle\right)+ i\sin(t \gamma B_0 \hbar/2)2\left(\frac{1+i}{4} |S_y;+ \rangle -\frac{1-i}{4}|S_y;-\rangle\right)$

so the probability is

$\cos^2 (t \gamma B_0 \hbar/2)$.

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