Principles of Quantum Mechanics · Problem 12.5.3

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-expectation-values

Problem

(1) Show that ⟨Jx⟩=⟨Jy⟩=0\langle J_x \rangle = \langle J_y \rangle = 0 in a state ∣jm⟩|jm\rangle. (2) Show that in these states ⟨Jx2⟩=⟨Jy2⟩=12ℏ2[j(j+1)−m2]\langle J_x^2 \rangle = \langle J_y^2 \rangle = \tfrac{1}{2}\hbar^2[j(j+1) - m^2] (use symmetry arguments to relate ⟨Jx2⟩\langle J_x^2 \rangle to ⟨Jy2⟩\langle J_y^2 \rangle). (3) Check that ΔJx⋅ΔJy\Delta J_x \cdot \Delta J_y from part (2) satisfies the inequality imposed by the uncertainty principle [Eq. (9.2.9)]. (4) Show that the uncertainty bound is saturated in the state ∣j,±j⟩|j, \pm j\rangle.

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