Electron spin in magnetic field

  • Thread starter fyzikapan
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  • #1
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Homework Statement


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.


Homework Equations


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

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


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.



The Attempt at a Solution

 

Answers and Replies

  • #2
nrqed
Science Advisor
Homework Helper
Gold Member
3,764
294
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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