Chapter 4

A beam of spin- \(\frac{1}{2}\) particles with speed \(v_{0}\) passes through a series of two SGz devices. The first SGz device transmits particles with \(S_{z}=\hbar / 2\) and filters out particles with \(S_{z}=-\hbar / 2\). The second SGz device transmits particles with \(S_{z}=-\hbar / 2\) and filters out particles with \(S_{z}=\hbar / 2\). Between the two devices is a region of length \(l_{0}\) in which there is a uniform magnetic field \(B_{0}\) pointing in the \(x\) direction. Determine the smallest value of \(l_{0}\) such that exactly 25 percent of the particles transmitted by the first SGz device are transmitted by the second device. Express your result in terms of \(\omega_{0}=e g B_{0} / 2 m c\) and \(v_{0}\).

A spin- \(\frac{1}{2}\) particle, initially in a state with \(S_{n}=\hbar / 2\) with \(\mathbf{n}=\sin \theta \mathbf{i}+\cos \theta \mathbf{k}\), is in a constant magnetic field \(B_{0}\) in the \(z\) direction. Determine the state of the particle at time \(t\) and determine how \(\left\langle S_{x}\right\rangle,\left\langle S_{y}\right\rangle\), and \(\left\langle S_{z}\right\rangle\) vary with time.

The matrix representation of the Hamiltonian for a photon propagating along the optic axis (taken to be the \(z\) axis) of a quartz crystal using the linear polarization states \(|x\rangle\) and \(|y\rangle\) as basis is given by $$ \hat{H} \underset{|x\rangle-(y) \text { basis }}{\longrightarrow}\left(\begin{array}{cc} 0 & -i E_{0} \\ i E_{0} & 0 \end{array}\right) $$ (a) What are the eigenstates and eigenvalues of the Hamiltonian? (b) A photon enters the crystal linearly polarized in the \(x\) direction, that is \(|\psi(0)\rangle=|x\rangle\). What is the \(|\psi(t)\rangle\), the state of the photon at time \(t\) ? Express your answer in the \(|x\rangle-|y\rangle\) basis. Show that the photon remains linearly polarized as it travels through the crystal. Explain what is happening to the polarization of the photon as time increases.

The lifetime of hydrogen in the \(2 p\) state to decay to the \(1 s\) ground state is \(1.6 \times 10^{-9} \mathrm{~s}\) [see (14.169)]. Estimate the uncertainty \(\Delta E\) in energy of this excited state. What is the corresponding linewidth in angstroms?