A rigid rotor of moment of inertia has energy eigenvalues

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch The postulates, measurement and collapse

Problem

A rigid rotor of moment of inertia

I

has energy eigenvalues

E_l = l(l+1)\hbar^2/(2I)

, with the

2l+1

states

|l,m\rangle

m=-l,\dots,l

, degenerate at each

E_l

. The rotor is prepared in the normalised state

|\psi\rangle = \tfrac12|2,-2\rangle + \tfrac{1}{\sqrt2}|2,0\rangle + \tfrac12|2,2\rangle,

where the three kets are mutually orthonormal eigenstates with

l=2

. ((a)) Write down

E_2

in terms of

\hbar

and

I

. ((b)) Compute the probability that a measurement of the rotor's energy returns

E_2

. ((c)) Explain, in one sentence, why this probability does not depend on the particular values

1/2,\,1/\sqrt2,\,1/2