Problem 1:

Two particles of mass m1 and m2 are separated by a fixed distance r.  Their center of mass is fixed at the origin of the coordinate system and they are free to rotate about their center of mass.  (The system is a "rigid rotator".)
(a)  Write down the Hamiltonian of the system.
(b)  Find the eigenvalues and eigenfunctions of this Hamiltonian.  What is the separation between adjacent levels?  What is the degeneracy of the eigenvalues?

Solution:

• Concepts:
The time-independent Schroedinger equation in three dimensions, the eigenfunctions of the orbital angular momentum operator.
• Reasoning:
We can write the Hamiltonian for the motion about the center of mass as that of a fictitious particle of reduced mass m moving in a central potential.  We solve the Schroedinger equation in spherical coordinates.
• Details of the calculation:
(a)  The Hamiltonian of the rotator is L2/(2I) where I is its moment of inertia,
I = m1r12 + m2r22 = μr2,  μ = m1m2/(m1 + m2).
(b) The eigenfunctions of H are the eigenfunctions of L2.
These are the spherical harmonics, Ylm(θ,φ) = <r|l,m>.
The eigenvalues are El = l(l + 1)ħ2/(2I).
It is customary to set  B = ħ/(4πI), and write El = Bhl(l + 1).
The separation between adjacent levels is  El - El-1 = Bh[l(l + 1) - (l - 1)l] = 2Bhl.
The separation increases linear with l.  Each energy eigenvalue is 2l + 1 fold degenerate.

Problem 2:

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:

• Concepts:
The eigenfunctions of the orbital angular momentum operator, the spherical harmonics
• Reasoning:
The common eigenfunctions of L2 and Lz are the spherical harmonics.  We have to write the given wave functions in terms of the spherical harmonics.
• Details of the calculation:
ψ(r) = (x + y + 3z)f(r) = (rsinθcosφ + rsinθsinφ - 3rcosθ)f(r)
= rf(r)(sinθcosφ + sinθsinφ - 3cosθ)
= rf(r)(sinθ ½(e + e-iφ) - i sinθ ½(e - e-iφ) - 3cosθ)
= rf(r)(sinθ ½e (1 - i) + sinθ ½ e-iφ (1 + i) - 3cosθ)
= rf(r)(2exp(-iπ/2) sinθe  + 2exp(iπ/2) sinθ e-iφ - 3cosθ).

Y1±1 = ∓(3/8π)1/2sinθ exp(±iφ),  Y10 = (3/4π)1/2cosθ.
ψ(r) = rf(r)(-exp(-iπ/2)(4π/3)1/2Y11(θ,φ) + exp(iπ/2)(4π/3)1/2Y1-1(θ,φ) - (12π)1/2Y10(θ,φ))
= rf(r)2√π(-exp(-iπ/2)(1/3)1/2Y11(θ,φ) + exp(iπ/2)(1/3)1/2Y1-1(θ,φ) - (3π)1/2Y10(θ,φ))
=  (44π/3)1/2rf(r)(-exp(-iπ/2)/111/2Y11(θ,φ) + exp(iπ/2)/111/2Y1-1(θ,φ) - 3/111/2Y10(θ,φ))
= R(r) (-exp(-iπ/2)/111/2Y11(θ,φ) + exp(iπ/2)/111/2Y1-1(θ,φ) - 3/111/2Y10(θ,φ)).
The function of angle is normalized.
ψ(r) is  an eigenfunction of L2 with eigenvalue 2ħ2, l = 1.

(b)  The possible values of ml are 1, 0, -1.

P(ml = 1) = 1/11,  P(ml = -1) = 1/11,  P(ml = 0) = 9/11.

(c)
The wave function ψklm(r,θ,φ) = Rkl(r)Ylm(θ,φ) = [ukl(r)/r]Ylm(θ,φ) is a product of a radial function Rkl(r) and the spherical harmonic Ylm(θ,φ).  The differential equation for ukl(r) is
[-(ħ2/(2m))(∂2/∂r2) + ħ2l(l+1)/(2mr2) + U(r)]ukl(r) = Eklukl(r).
Since ukl(r), Ekl and l are known, we can solve this equation for U(r).

#### Problem 4, solution

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(a)  If L2 is measure we can obtain with probability 4/9 and 0 with probability 5/9.
(b)  If Lz is measured we obtain 0 with probability 1.
(c)  y(r) is not an eigenfunction of L2. y(r) is an eigenfunction of Lz.