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Electron spin in magnetic field

  1. Jan 17, 2009 #1
    1. The problem statement, all variables and given/known data
    An electron is an eigenstate of sz at time t = 0 (spin up). It is in a magnetic field [tex]\vec B = (B \sin \theta, 0,B\cos\theta)[/tex]. Find the probability of finding the electron with spin down at time t.


    2. Relevant equations
    [tex]
    U(t) = \exp \left( -i \mathcal{H} t/\hbar\right)\\\\

    P(t) = \left| \left \langle \downarrow | \chi(t) \right \rangle \right|
    [/tex]


    3. The attempt at a solution
    [tex]
    \begin{align*}
    U(t) &= \exp \left( -i \mathcal{H} t/\hbar\right)\\
    &= \exp \left( -i \frac{et}{mc\hbar} \vec S \cdot \vec B\right)\\
    &= \exp \left( -i \frac{\omega_0 t}{2} \left(\sigma_x \sin \theta + \sigma_z \cos \theta\right)\right)\\
    &= \exp \left(-i \frac{\omega_0 t}{2} \left[ \begin{array}{cc}\cos \theta & \sin \theta \\
    \sin \theta & \cos \theta \end{array}\right]\right)\\
    &= \left( \begin{array}{cc}
    \exp (- i \frac{\omega_0 t}{2} \cos \theta) & \exp (- i \frac{\omega_0 t}{2} \sin \theta)\\
    \exp (- i \frac{\omega_0 t}{2} \sin \theta) & \exp ( i \frac{\omega_0 t}{2} \cos \theta)
    \end{array}\right)
    \end{align*}[/tex]

    Then at time t,
    [tex]
    \begin{align*}
    \chi(t) &= U(t) \left| \chi(0) \right \rangle\\
    &= \left( \begin{array}{cc}
    \exp (- i \frac{\omega_0 t}{2} \cos \theta) & \exp (- i \frac{\omega_0 t}{2} \sin \theta)\\
    \exp (- i \frac{\omega_0 t}{2} \sin \theta) & \exp ( i \frac{\omega_0 t}{2} \cos \theta)
    \end{array}\right)\left( \begin{array}{c}1\\0\end{array}\right)\\
    &= \left( \begin{array}{cc}
    \exp (- i \frac{\omega_0 t}{2} \cos \theta)\\
    \exp (- i \frac{\omega_0 t}{2} \sin \theta)\end{array}\right)
    \end{align*}
    [/tex]

    But this gives [tex]P(t) = \left| (\begin{array}{cc} 0 & 1 \end{array})\left( \begin{array}{cc}
    \exp (- i \frac{\omega_0 t}{2} \cos \theta)\\
    \exp (- i \frac{\omega_0 t}{2} \sin \theta)\end{array}\right)\right|^2 = 1[/tex], which is obviously wrong.

    I've looked over my math and don't see any obvious mistakes, so I'm not sure what I'm doing wrong.



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 18, 2009 #2

    nrqed

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    First, you have used for sigma_z the identity matrix. It should be 1 and -1 on the diagonal.
    Also, you seemed to assume that the exponential of a matrix is the matrix of the exponentials of the elements of the initial matrix. This is not true in general (works only for a diagonal matrix)
     
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