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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

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

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