**Problem 1:**

Two particles of mass m_{1} and m_{2} 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 L^{2}/(2I) where I is its moment of inertia,

I = m_{1}r_{1}^{2}+ m_{2}r_{2}^{2}= μr^{2}, μ = m_{1}m_{2}/(m_{1}+ m_{2}).

(b) The eigenfunctions of H are the eigenfunctions of L^{2}.

These are the spherical harmonics, Y_{lm}(θ,φ) = <**r**|l,m>.

The eigenvalues are E_{l}= l(l + 1)ħ^{2}/(2I).

It is customary to set B = ħ/(4πI), and write E_{l}= Bhl(l + 1).

The separation between adjacent levels is E_{l}- E_{l-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 L^{2}? If so, what is its l
value? If not, what are the possible values of l we may obtain when L^{2}
is measured?

(b) What are the probabilities for the particle to be found in various m_{l}
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 L^{2}and L_{z}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^{iφ}+ e^{-iφ}) - i sinθ ½(e^{iφ}- e^{-iφ}) - 3cosθ)

= rf(r)(sinθ ½e^{iφ}(1 - i) + sinθ ½ e^{-iφ}(1 + i) - 3cosθ)

= rf(r)(2^{-½}exp(-iπ/2)^{ }sinθe^{iφ}+ 2^{-½}exp(iπ/2) sinθ e^{-iφ}- 3cosθ).

Y_{1±1}= ∓(3/8π)^{1/2}sinθ exp(±iφ), Y_{10}= (3/4π)^{1/2}cosθ.

ψ(**r**) = rf(r)(-exp(-iπ/2)(4π/3)^{1/2}Y_{11}(θ,φ) + exp(iπ/2)(4π/3)^{1/2}Y_{1-1}(θ,φ) - (12π)^{1/2}Y_{10}(θ,φ))

= rf(r)2√π(-exp(-iπ/2)(1/3)^{1/2}Y_{11}(θ,φ) + exp(iπ/2)(1/3)^{1/2}Y_{1-1}(θ,φ) - (3π)^{1/2}Y_{10}(θ,φ))

= (44π/3)^{1/2}rf(r)(-exp(-iπ/2)/11^{1/2}Y_{11}(θ,φ) + exp(iπ/2)/11^{1/2}Y_{1-1}(θ,φ) - 3/11^{1/2}Y_{10}(θ,φ))

= R(r) (-exp(-iπ/2)/11^{1/2}Y_{11}(θ,φ) + exp(iπ/2)/11^{1/2}Y_{1-1}(θ,φ) - 3/11^{1/2}Y_{10}(θ,φ)).

The function of angle is normalized.

ψ(**r**) is an eigenfunction of L^{2}with eigenvalue 2ħ^{2}, l = 1.

(b) The possible values of m_{l}are 1, 0, -1.

P(m_{l}= 1) = 1/11, P(m_{l}= -1) = 1/11, P(m_{l}= 0) = 9/11.

(c) The wave function ψ_{klm}(r,θ,φ) = R_{kl}(r)Y_{lm}(θ,φ) = [u_{kl}(r)/r]Y_{lm}(θ,φ) is a product of a radial function R_{kl}(r) and the spherical harmonic Y_{lm}(θ,φ). The differential equation for u_{kl}(r) is

[-(ħ^{2}/(2m))(∂^{2}/∂r^{2}) + ħ^{2}l(l+1)/(2mr^{2}) + U(r)]u_{kl}(r) = E_{kl}u_{kl}(r).

Since u_{kl}(r), E_{kl}and l are known, we can solve this equation for U(r).

.

.

.

.

(a) If *L*^{2} is measure we can obtain
with probability 4/9 and 0 with
probability 5/9.

(b) If *L*_{z} is measured we obtain 0 with probability 1.

(c)
*y*(* r*) is not an eigenfunction of