Principles of Quantum Mechanics · Problem 12.4.2

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-three-dimensions

Problem

(1) Verify Eq. (12.4.2) by first constructing the

3 \times 3

matrices corresponding to

R(\varepsilon, \mathbf{i})

and

R(\varepsilon, \mathbf{j})

, to order

\varepsilon

. (2) Provide the steps connecting Eqs. (12.4.3) and (12.4.4a). (3) Verify that

L_x

and

L_y

defined in Eq. (12.4.1) satisfy Eq. (12.4.4a). The proof for other commutators follows by cyclic permutation.