A Modern Approach to Quantum Mechanics · Problem 4.14

A System of Two Spin-1/2 Particles — John S. Townsend

Physics — Quantum Mechanics · photon-polarization-quartz-crystal

Problem

The matrix representation of the Hamiltonian for a photon propagating along the optic axis (taken to be the zz axis) of a quartz crystal using the linear polarization states ∣x⟩|x\rangle and ∣y⟩|y\rangle as basis is given by H^→∣x⟩-∣y⟩ basis(0−iE0iE00)\hat{H} \xrightarrow{|x\rangle\text{-}|y\rangle\text{ basis}} \begin{pmatrix} 0 & -iE_0 \\ iE_0 & 0 \end{pmatrix} (a) What are the eigenstates and eigenvalues of the Hamiltonian? (b) A photon enters the crystal linearly polarized in the xx direction, that is ∣ψ(0)⟩=∣x⟩|\psi(0)\rangle = |x\rangle. What is the ∣ψ(t)⟩|\psi(t)\rangle, the state of the photon at time tt? Express your answer in the ∣x⟩|x\rangle-∣y⟩|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.

Browse the problem library