Principles of Quantum Mechanics · Problem 12.5.4

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · finite-rotations

Problem

(1) Argue that the eigenvalues of

J_x^{(j)}

and

J_y^{(j)}

are the same as those of

J_z^{(j)}

, namely

j\hbar, (j-1)\hbar, \ldots, (-j\hbar)

. Generalize the result to

\hat{\theta} \cdot \mathbf{J}^{(j)}

. (2) Show that

(J - j\hbar)[J - (j-1)\hbar][J-(j-2)\hbar]\cdots(J+j\hbar) = 0

where

J \equiv \hat{\theta} \cdot \mathbf{J}^{(j)}

. (Hint: In the case

J = J_z

what happens when both sides are applied to an arbitrary eigenket

|jm\rangle

? What about an arbitrary superposition of such kets?) (3) It follows from (2) that

J^{2j+1}

is a linear combination of

J^0, J^1, \ldots, J^{2j}

. Argue the same goes for

J^{2j+k}

k = 1, 2, \ldots