Orbital angular momentum

Problem:

Consider a spinless particle of mass m.  What values can the magnitude of the angular momentum J take in quantum mechanics when it is measured?

Solution:

Problem:

The magnitude of the orbital angular momentum L of a hydrogen atom is found to be 30½ħ.  Lz is measured and found to be 3ħ.
(a)  Which values of the principle quantum number n are consistent with these measurements?
(b)  What is the value of Lx2 + Ly2?

Solution:

Problem:

Show that a state with position wave function (y - iz)k is an eigenstate of the orbital angular momentum operator Lx and find its eigenvalue.  Are there conditions on k?

Solution:

Problem:

For a simple particle moving in space, show that the wave function ψlm(r) = x2 + y2 - 2z2 represents a simultaneous eigenstate of L2 and Lz with eigenvalues l(l + 1)ħ2 and mħ.  Determine l and m.  Find a function with the same eigenvalue for L2 and the maximum possible eigenvalue for Lz.

Solution:

Problem:

The wave function of a particle subjected to a spherically symmetric potential U(r) is given by ψ(r) = (x + y + 3z)f(r).
(a)  Is ψ an eigenfunction of L2?  If so, what is its l value?  If not, what are the possible values of l we may obtain when L2 is measured?
(b)  What are the probabilities for the particle to be found in various ml states?  
(c)  Suppose it is known somehow that ψ(r) is an energy eigenfunction with eigenvalue E.  Indicate how we may find U(r).

Solution:

Problem:

The ladder operator L+ is given by
L+ = ħ exp(iφ) [∂/∂θ + i cotθ ∂/∂φ].
The spherical harmonics are given by Ylm(θ,φ) = Flm(θ)exp(imφ).
(a)  Show that Yll(θ,φ) = cl sinl(θ)exp(ilφ), where cl is a constant.
(b)  Indicate how you would proceed to find the θ-dependence of Ylm(θ,φ).

Solution:

Problem:

A particle is known to be in an eigenstate of L2 and Lz.  Prove that the expectation values satisfy
<Lx> = <Ly> = 0,   <Lx2> = <Ly2> = (l(l+1)ħ2 - m2ħ2)/2.

Solution:

The rigid rotator