time evolution of spin state

noospace

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 [latex]U = e^{-i/\hbar \hat{H} t}[/latex]

3. The attempt at a solution

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

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

so the time evolved state vector is

[latex]|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)[/latex]

[latex]|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) [/latex]

so the probability is

[latex]\cos^2 (t \gamma B_0 \hbar/2)[/latex].