Principles of Quantum Mechanics · Problem 12.4.4

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · vector-operators

Problem

Recall that

\mathbf{V}

is a vector operator if its components

V_i

transform as

U^\dagger[R] V_i U[R] = \sum_j R_{ij} V_j

(1) For an infinitesimal rotation

\delta\mathbf{\theta}

, show, on the basis of the previous exercise, that

\sum_j R_{ij} V_j = V_i + (\delta\mathbf{\theta} \times \mathbf{V})_i = V_i + \sum_j \sum_k \varepsilon_{ijk}(\delta\theta)_j V_k

(2) Feed in

U[R] = 1 - (i/\hbar)\delta\mathbf{\theta} \cdot \mathbf{L}

into the left-hand side of Eq. (12.4.13) and deduce that

[V_i, L_j] = i\hbar \sum_k \varepsilon_{ijk} V_k