Principles of Quantum Mechanics · Problem 1.8.3

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · eigenvalue-problem-hermitian

Problem

Consider the Hermitian matrix

\Omega = \frac{1}{2}\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & -1 \\ 0 & -1 & 3 \end{bmatrix}

(1) Show that

\omega_1 = \omega_2 = 1

;

\omega_3 = 2

. (2) Show that

|\omega = 2\rangle

is any vector of the form

\frac{1}{(2a^2)^{1/2}}\begin{bmatrix} 0 \\ a \\ -a \end{bmatrix}

(3) Show that the

\omega = 1

eigenspace contains all vectors of the form

\frac{1}{(b^2 + 2c^2)^{1/2}}\begin{bmatrix} b \\ c \\ c \end{bmatrix}

either by feeding

\omega = 1

into the equations or by requiring that the

\omega = 1

eigenspace be orthogonal to

|\omega = 2\rangle