A rigid diatomic molecule has rotational angular momentum and is prepared in the…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

A rigid diatomic molecule has rotational angular momentum

J=1

and is prepared in the state

|1,-1\rangle

(eigenstate of

\hat J_z

with eigenvalue

-\hbar

). It is rotated by

\beta = \pi/3

about the

y

axis. The relevant Wigner

d^{(1)}

matrix elements are

d^{(1)}_{1,-1}(\beta) = \frac{1-\cos\beta}{2},\quad d^{(1)}_{0,-1}(\beta) = -\frac{\sin\beta}{\sqrt2},\quad d^{(1)}_{-1,-1}(\beta) = \frac{1+\cos\beta}{2}.

((a)) Give the amplitudes for finding

J_z = +\hbar

0

-\hbar

after the rotation. ((b)) Give the corresponding probabilities.