A Modern Approach to Quantum Mechanics · Problem 1.8

Rotation of Basis States and Matrix Mechanics — John S. Townsend

Physics — Quantum Mechanics · expectation-value-uncertainty-sz

Problem

The state of a spin-

\frac{1}{2}

particle is given by

|\psi\rangle = \frac{i}{\sqrt{3}}|{+}\mathbf{z}\rangle + \sqrt{\frac{2}{3}}|{-}\mathbf{z}\rangle

What are

\langle S_z\rangle

and

\Delta S_z

for this state? Suppose that an experiment is carried out on 100 particles, each of which is in this state. Make up a reasonable set of data for

S_z

that could result from such an experiment. What if the measurements were carried out on 1,000 particles? What about 10,000?