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, Y_lm(θ,φ) = <r|l,m>.
  The eigenvalues are E_l = l(l + 1)ħ²/(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 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 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 L² 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^-½exp(-iπ/2) sinθe^iφ  + 2^-½exp(iπ/2) sinθ e^-iφ - 3cosθ).
  
  Y_1±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)^1/2Y_1-1(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)^1/2Y_1-1(θ,φ) - (3π)^1/2Y₁₀(θ,φ))
  =  (44π/3)^1/2rf(r)(-exp(-iπ/2)/11^1/2Y₁₁(θ,φ) + exp(iπ/2)/11^1/2Y_1-1(θ,φ) - 3/11^1/2Y₁₀(θ,φ))
  = R(r) (-exp(-iπ/2)/11^1/2Y₁₁(θ,φ) + exp(iπ/2)/11^1/2Y_1-1(θ,φ) - 3/11^1/2Y₁₀(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 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
  [-(ħ²/(2m))(∂²/∂r²) + ħ²l(l+1)/(2mr²) + U(r)]u_kl(r) = E_klu_kl(r).
  Since u_kl(r), E_kl and l are known, we can solve this equation for U(r).

Problem 3, solution

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 L_z is measured we obtain 0 with probability 1.
(c)  y(r) is not an eigenfunction of L². y(r) is an eigenfunction of L_z.