A Modern Approach to Quantum Mechanics · Problem 1.6

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

Physics — Quantum Mechanics · spin-half-orthogonality

Problem

Show that the state

|{-}\mathbf{n}\rangle = \sin\frac{\theta}{2}|{+}\mathbf{z}\rangle - e^{i\phi}\cos\frac{\theta}{2}|{-}\mathbf{z}\rangle

satisfies

\langle{+}\mathbf{n}|{-}\mathbf{n}\rangle = 0

, where the state

|{+}\mathbf{n}\rangle

is given in Problem 1.3. Verify that

\langle{-}\mathbf{n}|{-}\mathbf{n}\rangle = 1