A Modern Approach to Quantum Mechanics · Problem 9.20

Bound States of Central Potentials — John S. Townsend

Physics — Quantum Mechanics · rigid-rotator-wave-function-time-evolution

Problem

The wave function of a rigid rotator with a Hamiltonian H^=L^2/2I\hat{H} = \hat{L}^2/2I is given by ⟨θ,ϕ∣ψ(0)⟩=34πsin⁡θsin⁡ϕ\langle\theta, \phi|\psi(0)\rangle = \sqrt{\frac{3}{4\pi}}\sin\theta\sin\phi (a) What is ⟨θ,ϕ∣ψ(t)⟩\langle\theta, \phi|\psi(t)\rangle? Suggestion: Express the wave function in terms of the Yl,mY_{l,m}'s. (b) What values of LzL_z will be obtained if a measurement is carried out and with what probability will these values occur? (c) What is ⟨Lx⟩\langle L_x\rangle for this state? Suggestion: Use bra-ket notation and express the operator L^x\hat{L}_x in terms of raising and lowering operators. (d) If a measurement of LxL_x is carried out, what result(s) will be obtained? With what probability? Suggestion: If you have worked out Problem 3.15, you can take good advantage of the expressions for the states ∣1,m⟩x|1, m\rangle_x.

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