A Modern Approach to Quantum Mechanics · Problem 3.5

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · rotation-operator-spin-half

Problem

This problem demonstrates another way (also see Problem 3.2) to determine the eigenstates of

\hat{S}_n = \hat{\mathbf{S}} \cdot \mathbf{n}

. The operator

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

rotates spin states by an angle

\theta

counterclockwise about the

y

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

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

Suggestion: Use the states

|{+}\mathbf{z}\rangle

and

|{-}\mathbf{z}\rangle

as a basis. Express the operator

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

in matrix form by expanding

\hat{R}

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

\hat{S}_y^2

\hat{S}_y^3

, and so on. ((b)) Apply

\hat{R}

in matrix form to the state

|{+}\mathbf{z}\rangle

to obtain the state

|{+}\mathbf{n}\rangle

given in Problem 3.2 with

\phi = 0

, that is, rotated by angle

\theta

in the

x

z

plane. Show that

\hat{R}|{-}\mathbf{z}\rangle

differs from

|{-}\mathbf{n}\rangle

by an overall phase.