A Modern Approach to Quantum Mechanics · Problem 3.12

Time Evolution — John S. Townsend

Physics — Quantum Mechanics · spin-half-operators-outer-product

Problem

Verify for a spin-

\frac{1}{2}

particle that (a)

\hat{S}_z = (\hbar/2)|{+}\mathbf{z}\rangle\langle{+}\mathbf{z}| - (\hbar/2)|{-}\mathbf{z}\rangle\langle{-}\mathbf{z}|

and (b) the raising and lowering operators may be expressed as

\hat{S}_+ = \hbar|{+}\mathbf{z}\rangle\langle{-}\mathbf{z}| \quad \text{and} \quad \hat{S}_- = \hbar|{-}\mathbf{z}\rangle\langle{+}\mathbf{z}|

Note: It is sufficient to examine the action of these operators on the basis states

|{+}\mathbf{z}\rangle

and

|{-}\mathbf{z}\rangle

, which of course form a complete set.