Principles of Quantum Mechanics · Problem 15.3.1

Addition of Angular Momenta — R. Shankar

Physics — Quantum Mechanics · irreducible-tensor-operators

Problem

(1) Show that Eq. (15.3.11) follows from Eq. (15.3.10) when one considers infinitesimal rotations. (Hint:

D^{(k)}_{q'q} = \langle kq'|I - (i\delta\boldsymbol{\theta}\cdot\mathbf{J})/\hbar|kq\rangle

. Pick

\delta\boldsymbol{\theta}

along, say, the

x

direction and then generalize the result to the other directions.) (2) Verify that the spherical tensor

V_1^q

constructed out of

\mathbf{V}

as in Eq. (15.3.15) obeys Eq. (15.3.11).