Problem 1:

A particle with mass m and with zero energy has a time-independent wave function
ψ(x) = x exp(-x²/L²).  Determine the potential energy U(x) of the particle.

Solution:

• Concepts:
  The time-independent Schroedinger equation
• Reasoning:
  If ψ(x) and E are given, we can solve for U(x).
• Details of the calculation:
  The normalization does not matter for this problem.  Set A = 1.
  ψ(x) = x exp(-x²/L²).
  Schroedinger equation in one dimension:  -(ħ²/(2m)) d²ψ(x)/dx² + U(x) ψ(x) = E ψ(x).
  Here E = 0, d²ψ(x)/dx² = (2m/ ħ²)U(x) ψ(x).
  dψ(x)/dx = exp(-x²/L²) - 2x² exp(-x²/L²)/L².
  d²ψ(x)/dx² = - 2x exp(-x²/L²)/L² -  4x exp(-x²/L²)/L² + 4x³ exp(-x²/L²)/L⁴.
  -6x exp(-x²/L²)/L² + 4x³ exp(-x²/L²)/L⁴ = (2m/ħ²)U(x) x exp(-x²/L²).
  -6/L² + 4x²/L⁴ = (2m/ħ²)U(x),  U(x) = -3 ħ²/(mL²) + 2ħ²x²/(mL⁴).
  
  Or just recognize the given wavefunction as the 1^st excited state of the harmonic oscillator.
  Φ₁(x) = x exp(-½mωx²/ħ) (not normalized).
  L² = 2 ħ/(mω).  The potential energy is ½mω²x² - (3/2) ħω so that E = 0.

Problem 2:

A quantum mechanical system can be described by a Hamiltonian

where  a is a real number, in the orthonormal basis |1> and |2>.

(a)  What are the energy eigenvalues and normalized eigenvectors of this Hamiltonian?
(b)  What is the probability for finding the system in state |2> at time t > 0 if the system is known to be in state |1> at t = 0?

Solution:

• Concepts:
  The fundamental assumptions of quantum mechanics, the evolution operator
• Reasoning:
  The system is not in a stationary state. We need to find the eigenstates of H and expand the state in terms of these eigenstates.
• Details of the calculation:
  We are given the matrix of the Hermitian operator H in some basis.  To find the eigenvalues E we set the determinant of the matrix (H - EI) equal to zero and solve for E.  To find the corresponding eigenvectors {|α>}, we substitute each eigenvalue E back into the equation (H - EI)|α> and solve for the expansion coefficients of |α> in the given basis.
  The evolution operator is a unitary operator defined through |ψ(t)> = U(t,t₀)| ψ(t₀)>.
  If H does not explicitly depend on time, then U(t,t₀) = exp(-(i/ħ)H(t-t₀)).
  If |ψ(t₀)> is an eigenfunction of H with eigenvalue E, then |ψ(t)> =  exp(-(i/ħ)E(t-t₀))|ψ(t₀)>.
  (a)  The eigenvalues of H are E, where
  
  
  
  E² - a ² = 0,  E = ±a.
  The normalized eigenstates of H are |±> =  A₁|1> + A₂|2>.
  For E = a we have -aA₁ + aA₂ = 0, A₂ = A₁. 
  |+> = 2^-½|1> + 2^-½| 2>.
  For E = -a we have aA₁ + aA₂ = 0,  A₂ =- A₁
  |-> = 2^-½|1> - 2^-½| 2>.
  
  (b)  |ψ(t)> = exp(-iHt/ħ)|1>.  P₂(t) = |<2|ψ(t)>|².
  |1> = 2^-½(|+> + |->),  |2> = 2^-½(|+> - |->).
  Global phase factors are irrelevant.
  
  |ψ(t)> = 2^-½(exp(-iat/ħ)|+> + exp(iat/ħ)|->). 
  <2|ψ(t)> = ½(exp(-iat/ħ) - exp(iat/ħ))= -i sin(|at/ħ).
  P₁(t) = sin²(at/ħ).

Problem 3:

You prepare an ensemble of identical experiments at t = 0.  You measure the values of a physical observable A at time t = t₁ > 0 in all of the experiments of the ensemble.  At time t₂ > t₁ you measure observable A again in all of the experiments - i.e., you follow the measurement at t = t₁ in each experiment with a subsequent measurement of A in each experiment at t = t₂.

True or False?  Explain!
(a)  The values you obtain for A across all of the experiments at time t = t₁ are the same given you have made all of the measurements at the same time t = t₁.
(b)  The value you obtain for A in each experiment at t = t₂ may be different relative to the value obtained for A in the same experiment at t = t₁.
(c)  The expectation value of A is the same at both t = t₁ and t = t₂.
(d)  The values you obtain for A in any of the experiments at either t = t₁ or t = t₂ must be one of the eigenvalues of its associated Hermitian operator.

At t = t₃ > t₂ you measure a second physical observable B across all of the experiments - i.e., you follow the measurement of A at t = t₂ in each experiment with a measurement of B at t = t₃ in each experiment. The commutator of the two Hermitian operators associated with A and B satisfies the following: [A, B] ≠ 0.

True or False?  Explain!
(e)  The values you obtained for A at t = t₂ in any of these experiments will remain the same at t = t₃ given you are measuring a different physical quantity B, not A.

Solution:

• Concepts:
  Fundamental assumptions of QM, the evolution operator, commuting observables
• Reasoning:
  The expectation value of an observable A is constant in time if A and H commute.
• Details of the calculation:
  (a)  This statement is false in general.
  This statement is only true for an arbitrary time t₁ if [A,H] = 0 and the systems were prepared in an eigenstate of A at t = 0.
  
  (b)  True.  Only if [A,H] = 0 is the probability of measuring the same value for A at t₁ and t₂ equal to 1.
  
  (c)  This statement is false in general.
  For the expectation value of the operator A we write
  (d/dt)<Ψ(t)|A|Ψ(t)>
  = [(∂/∂t)<Ψ(t)|]A|Ψ(t)> + <Ψ(t)|∂A/∂t|Ψ(t)> + <Ψ(t)|A[(∂/∂t)|Ψ(t)>]
  = <Ψ(t)|∂A/∂t|Ψ(t)> + (iħ)^-1<Ψ(t)|AH - HA|Ψ(t)>
  = <Ψ(t)|∂A/∂t|Ψ(t)> + (iħ)^-1<Ψ(t)|[A,H]|Ψ(t)>.
  Here we have used the Schroedinger equation,
  (iħ∂/∂t)|Ψ> = H|Ψ>,  (-iħ∂/∂t)<Ψ| = H<Ψ|.
  If [A,H] ≠  0, the expectation value of A changes with time.
  
  (d)  True.  A measurement of A will always yield one of the eigenvalues of A.
  
  (e)  False.   Since [A, B] ≠ 0, a measurement of B will put the system into an eigenstate of B, and destroy the exact information about the value of A.

Problem 4:

In the Aharanov Bohm experiment, a beam of electrons is split in two and they pass on opposite sides of a very long solenoid of radius r_a  <  r.  The solenoid is centered at the origin of the x-y plane, is pointing along the z-axis, and is carrying a steady current I.  The solenoid is infinitely long and the magnetic field inside is uniform.  The magnetic field outside the solenoid is zero.  The two beams (wave functions) acquire opposite phase shifts as they pass the solenoid along opposite sides on their way to the detector, resulting in an interference pattern.  In this problem, we will calculate these phase shifts.

(a)  Adopting the Coulomb gauge, show that the vector potential A for r > r_a is given by

where φ is the azimuthal angle in the x-y plane and Φ is the magnetic flux through the solenoid.

(b)  The classical Hamiltonian for a particle of charge q and momentum p in the presence of an electromagnetic field is given by

where A is the vector potential and V the scalar potential.
Write down the time-dependent Schroedinger equation for a quantum particle of charge q and wave function Ψ in the presence of an electromagnetic field.

(c )  Let's define a new wave function

where r₀ is some arbitrarily chosen reference point.  Using this definition, show that the Schroedinger equation for the new wave function Ψ' reduces to a usual Schroedinger equation without the vector potential|A|.

(d)  Back to the experiment.  Before exiting the entrance slits, the beam is moving through a region with A = 0.  Using the vector potential found in (a), show that the wave functions acquire a phase g = ±qΦ(2ħ), resulting in a (measurable) phase difference of (q/ħ)Φ as electrons pass the solenoid on opposite sides although the B field itself is zero.

(e)   Based on these observations, does the vector potential have a physical meaning and is the quantum theory gauge invariant?

Solution:

• Concepts:
  The vector potential A in magnetostatics, the Schroedinger equation
• Reasoning:
  In the presence of a magnetic field, the Hamiltonian H(r), and therefore the wave function Ψ(r), will depend on the vector potential A.
• Details of the calculation: