Exercises

1. Find the Pauli representations of $S_x$, $S_y$, and $S_z$ for a spin-1 particle.

2. Find the Pauli representations of the normalized eigenstates of $S_x$ and $S_y$ for a spin-$1/2$ particle.
3. Suppose that a spin-$1/2$ particle has a spin vector which lies in the $x$-$z$ plane, making an angle $\theta$ with the $z$-axis. Demonstrate that a measurement of $S_z$ yields $\hbar/2$ with probability $\cos^2(\theta/2)$, and $-\hbar/2$ with probability $\sin^2(\theta/2)$.

4. An electron is in the spin-state
   
   $$\chi = A \left(\begin{array}{c}1-2 {\rm i}\ 2\end{array}\right)$$
   
   
   in the Pauli representation. Determine the constant $A$ by normalizing $\chi$. If a measurement of $S_z$ is made, what values will be obtained, and with what probabilities? What is the expectation value of $S_z$? Repeat the above calculations for $S_x$ and $S_y$.

5. Consider a spin-$1/2$ system represented by the normalized spinor
   
   $$\chi =\left(\begin{array}{c}\cos\alpha\ \sin\alpha \exp( {\rm i} \beta)\end{array}\right)$$
   
   
   in the Pauli representation, where $\alpha$ and $\beta$ are real. What is the probability that a measurement of $S_y$ yields $-\hbar/2$?

6. An electron is at rest in an oscillating magnetic field
   
   $${\bf B} = B_0 \cos(\omega t) {\bf e}_z,$$
   
   
   where $B_0$ and $\omega$ are real positive constants.
   1. Find the Hamiltonian of the system.
   2. If the electron starts in the spin-up state with respect to the $x$-axis, determine the spinor $\chi(t)$ which represents the state of the system in the Pauli representation at all subsequent times.
   3. Find the probability that a measurement of $S_x$ yields the result $-\hbar/2$ as a function of time.
   4. What is the minimum value of $B_0$ required to force a complete flip in $S_x$?