A Modern Approach to Quantum Mechanics · Problem 5.16

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · density-matrix-pure-state-spin-half

Problem

A spin-12\frac{1}{2} particle is in the pure state ∣ψ⟩=a∣+z⟩+b∣−z⟩|\psi\rangle = a|{+z}\rangle + b|{-z}\rangle. (a) Construct the density matrix in the SzS_z basis for this state. (b) Starting with your result in (a), determine the density matrix in the SxS_x basis, where ∣+x⟩=12∣+z⟩+12∣−z⟩∣−x⟩=12∣+z⟩−12∣−z⟩|{+x}\rangle = \frac{1}{\sqrt{2}}|{+z}\rangle + \frac{1}{\sqrt{2}}|{-z}\rangle \qquad |{-x}\rangle = \frac{1}{\sqrt{2}}|{+z}\rangle - \frac{1}{\sqrt{2}}|{-z}\rangle (c) Use your result for the density matrix in (b) to determine the probability that a measurement of SxS_x yields ℏ/2\hbar/2 for the state ∣ψ⟩|\psi\rangle.

Related theory

Browse the problem library