Principles of Quantum Mechanics · Problem 1.8.5

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-unitary

Problem

Consider the matrix

\Omega = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}

(1) Show that it is unitary. (2) Show that its eigenvalues are

e^{i\theta}

and

e^{-i\theta}

. (3) Find the corresponding eigenvectors; show that they are orthogonal. (4) Verify that

U^\dagger\Omega U =

(diagonal matrix), where

U

is the matrix of eigenvectors of

\Omega