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Quantum Physics II
Exercise # 1
2019/3/6
1. A system is in the |ℓ, 𝑚⟩ eigenstate of 𝐿2 , 𝐿𝑧 .
(a) Show that the expectation values of 𝐿± = 𝐿𝑥 ± 𝐿𝑦 , 𝐿𝑥 , and 𝐿𝑦 are all vanish.
(b) Determine the uncertainty relation ∆𝐿𝑥 ∆𝐿𝑦 for the cases 𝑚 = 0 and 𝑚 = ℓ.
2. What if we like x instead of z? Assume that 𝜓(𝜃, 𝜙) is the eigenfunction of 𝐿2 and 𝐿𝑥 with
eigenvalues 2ℏ and ℏ, respectively.
(a) Express the 𝜓(𝜃, 𝜙) as a linear combination of eigenfunctions of 𝐿2 and 𝐿𝑧 .
(b) Write explicitly 𝜓(𝜃, 𝜙) as the function of the two variables, 𝜃 and 𝜙.
3. Assume 𝑥± = 𝑥 ± 𝑖𝑦.
(a) Calculate the commutator: [𝐿𝑧 , 𝑥± ].
(b) Show that 𝑥+ |ℓ, ℓ⟩ is an eigenstate of 𝐿𝑧 with the eigenvalue ℏ(ℓ + 1).
4. A particle of mass M is constrained to move on a spherical surface of radius a.
(a) What is the Hamiltonian of the system?
(b) What are the energy levels and degeneracies?
(c) What are the wavefunctions of the energy eigenstates?
5. At a given instant of time, a rigid rotator is in the state
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𝜓(𝜃, 𝜙) = √4𝜋 sin(𝜙)sin(𝜃) .
(a) Determine the values of 𝐿𝑧 found when measurement is done?
(b) What are the probabilities of finding the values obtained in (a)?
(c) What is the expectation value, ⟨𝐿𝑧 ⟩, for this state?
(d) What is the expectation value, ⟨𝐿2 ⟩, for this state?