A diatomic molecule's rotational angular momentum has quantum number
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 diatomic molecule's rotational angular momentum has quantum number J=1. It is prepared, along a laboratory z-axis fixed by an external field, in the state ∣J=1,M=−1⟩.A first apparatus applies the rotation R^(y^,β1) with β1=45∘ about the y-axis to the molecule's angular momentum state (the field direction, and hence the axis along which M is subsequently measured, is unchanged). Immediately afterwards, M is measured along the original z-axis.
((a)) Using the Wigner d(1) matrix, write the state after the first rotation as a superposition of ∣1,1⟩, ∣1,0⟩, ∣1,−1⟩, and give the probability that this first measurement yields M=0.
((b)) Given that the first measurement yields M=0, state the normalised angular momentum state immediately after that measurement.
((c)) A second apparatus now applies R^(y^,β2) with β2=135∘ about the y-axis to this collapsed state, and M is measured a second time along the same z-axis. Find the probability that this second measurement yields M=+1.
((d)) Find the overall probability that the sequence of two measurements yields, in order, M=0 and then M=+1.
((e)) Now suppose the first measurement is made but its result is not recorded. Find the probability that the second measurement yields M=+1. Compare it with the probability when no first measurement is made at all, so that both rotations act in succession on ∣1,−1⟩, and say what the first measurement changed.