9.E: Spin Angular Momentum (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 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 $\sin^2(\theta/2)$.
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 previous 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 $S_x$?

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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