Principles of Quantum Mechanics · Problem 12.4.3

Rotational Invariance and Angular Momentum — R. Shankar

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

Problem

We would like to show that

\hat{\theta} \cdot \mathbf{L}

generates rotations about the axis parallel to

\hat{\theta}

. Let

\delta\theta

be an infinitesimal rotation parallel to

\mathbf{\theta}

. (1) Show that when a vector

\mathbf{r}

is rotated by an angle

\delta\theta

, it changes to

\mathbf{r} + \delta\mathbf{\theta} \times \mathbf{r}

. (It might help to start with

\mathbf{r} \perp \delta\mathbf{\theta}

and then generalize.) (2) We therefore demand that (to first order, as usual)

\psi(\mathbf{r}) \underset{U[R(\delta\theta)]}{\longrightarrow} \psi(\mathbf{r} - \delta\mathbf{\theta} \times \mathbf{r}) = \psi(\mathbf{r}) - (\delta\mathbf{\theta} \times \mathbf{r}) \cdot \nabla\psi

Comparing to

U[R(\delta\theta)] = I - (i\,\delta\theta/\hbar) L_{\hat{\theta}}

, show that

L_{\hat{\theta}} = \hat{\theta} \cdot \mathbf{L}