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Time evolution of spin state

  1. Mar 16, 2008 #1
    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 [itex]U = e^{-i/\hbar \hat{H} t}[/itex]

    3. The attempt at a solution

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

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

    so the time evolved state vector is

    [itex]|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)[/itex]

    [itex]|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) [/itex]

    so the probability is

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