Quantum Mechanics: Concepts and Applications · Problem 5.30

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · pauli-matrices-properties

Problem

Consider the Pauli matrices

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

(a) Verify that

\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I

, where

I

is the unit matrix

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

(b) Calculate the commutators

[\sigma_x,\ \sigma_y]

[\sigma_x,\ \sigma_z]

, and

[\sigma_y,\ \sigma_z]

. (c) Calculate the anticommutator

\sigma_x\sigma_y + \sigma_y\sigma_x

. (d) Show that

e^{i\theta\sigma_y} = I\cos\theta + i\sigma_y\sin\theta

, where

I

is the unit matrix. (e) Derive an expression for

e^{i\theta\sigma_z}

by analogy with the one for

\sigma_y