Electron spin in magnetic field

[fyzikapan]

Homework Statement

An electron is an eigenstate of s_z at time t = 0 (spin up). It is in a magnetic field $$\vec B = (B \sin \theta, 0,B\cos\theta)$$. Find the probability of finding the electron with spin down at time t.

Homework Equations

$$U(t) = \exp \left( -i \mathcal{H} t/\hbar\right)\\\\ P(t) = \left| \left \langle \downarrow | \chi(t) \right \rangle \right|$$

The Attempt at a Solution

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

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

But this gives $$P(t) = \left| (\begin{array}{cc} 0 & 1 \end{array})\left( \begin{array}{cc} \exp (- i \frac{\omega_0 t}{2} \cos \theta)\\ \exp (- i \frac{\omega_0 t}{2} \sin \theta)\end{array}\right)\right|^2 = 1$$, 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

 

[nrqed]

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)