Principles of Quantum Mechanics · Problem 1.8.8

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · hermitian-matrices-anticommutation

Problem

Consider Hermitian matrices

M^1

M^2

M^3

M^4

that obey

M^i M^j + M^j M^i = 2\delta^{ij}I, \qquad i, j = 1, \ldots, 4

(1) Show that the eigenvalues of

M^i

are

\pm 1

. (Hint: go to the eigenbasis of

M^i

, and use the equation for

i = j

.) (2) By considering the relation

M^i M^j = -M^j M^i \quad \text{for } i \neq j

show that

M^i

are traceless. [Hint:

\text{Tr}(ACB) = \text{Tr}(CBA)

.] (3) Show that they cannot be odd-dimensional matrices.