Exercises

1. Demonstrate that the operators defined in Equations (427)-(429) are Hermitian, and satisfy the commutation relations (417).

2. Prove the Baker-Hausdorff lemma, (447).

3. Find the Pauli representations of the normalized eigenstates of $S_x$ and $S_y$ for a spin-$1/2$ particle.

4. Suppose that a spin-$1/2$ particle has a spin vector that 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 $\sin^2(\theta/2)$ .

5. 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 $S_y$ .

6. 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 $-\hbar/2$ ?

7. 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 $\chi(t)$ that 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 $S_x$ ?