Principles of Quantum Mechanics · Problem 12.5.13

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-measurement-probabilities

Problem

Consider a particle in a state described by

\psi = N(x + y + 2z)\, e^{-ar}

where

N

is a normalization factor. (1) Show, by rewriting the

Y_1^{\pm 1, 0}

functions in terms of

x

y

z

, and

r

, that

Y_1^{\pm 1} = \mp\left(\frac{3}{4\pi}\right)^{1/2}\frac{x \pm iy}{2^{1/2}r}

Y_1^0 = \left(\frac{3}{4\pi}\right)^{1/2}\frac{z}{r}

(2) Using this result, show that for a particle described by

\psi

above,

P(l_z = 0) = \frac{2}{3}

P(l_z = +\hbar) = 1/6 = P(l_z = -\hbar)