A Modern Approach to Quantum Mechanics · Problem 5.14

Wave Mechanics in One Dimension — John S. Townsend

Physics — Quantum Mechanics · rotation-operator-teleportation-bell-state

Problem

The first part of this problem follows the procedure outlined in Problem 3.5. The operator

\hat{R}(\theta\mathbf{i}) = e^{-i\hat{S}_x\theta/\hbar}

rotates spin states by an angle

\theta

counterclockwise about the

x

axis. (a) Show that this rotation operator can be expressed in the form

\hat{R}(\theta\mathbf{i}) = \cos\frac{\theta}{2} - \frac{2i}{\hbar}\hat{S}_x\sin\frac{\theta}{2}

Suggestion: Use the states

|{+z}\rangle

and

|{-z}\rangle

as a basis. Express the operator

\hat{R}(\theta\mathbf{i})

in matrix form by expanding

\hat{R}

in a Taylor series. Examine the explicit form for the matrices representing

\hat{S}_x^2

\hat{S}_x^3

, and so on. (b) Use matrix mechanics to determine which of the results

|\Psi^{(+)}_{12}\rangle

|\Phi^{(-)}_{12}\rangle

of Alice's Bell-state measurement yields a state for Bob's particle that is rotated into the state

|\psi\rangle

by the operator

\hat{R}(\pi\mathbf{i})