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Physics 422 (Opportunity II)
March 14, 2001
Make sure your name is on all pages of your test. Show all work clearly!!!! The question is
wrong if all work is not clearly shown. It is your responsibility to make sure you understand the question
before attempting it. Assume ideal conditions unless otherwise stated. Label things clearly. The final
answer must include units where appropriate.
Potentially useful information.
Raising and lowering operators:
aˆk , 
 
i

 qˆk ,  pˆ k , 
2 


Schrödinger eq. u(r) where u(r) = R(r) r
aˆ †k , 
 
i

 qˆk ,  pˆ k , 
2 



d2
(  1)



2
2 r 2
 2 dr
2

 V (r )  u (r )  Eu (r )

1) Consider S-wave scattering ( = 0) from a finite potential “bump”.
Vo r < a
V=
0
r >a
Assume low incident energies E but E>Vo
a) Write down the Schrödinger eigenvalue equation in terms of u = Rr for r<a.
b) Write down the Schrödinger eigenvalue equation in terms of u = Rr for r>a.
c) Use the definition of ko given in equation 1, and write down the solution for r<a and that satisfies the
proper boundary condition at r=0.
k0 
2
2
( E  V0 )
(Equation 1)
k 
2
2
E
(Equation 2)
d) Show explicitly that the solution given in part c is indeed the solution.
e) Use the definition of k given in equation 2, and write down the phase shifted (o) solution for r>a.
f) Show explicitly that the solution given in part e is indeed the solution.
g) By applying the appropriate boundary conditions, derive an expression that relates k, ko, o and a.
h) What is the total scattering cross section, in terms of k, ko, o and a? (assume (ka)/(koa) <<1).Also,
assume not resonance phenomena.
2) As the name implies, a Superconducting Quantum Interference Device (SQUID) utilizes quantum
interference to measure very small magnetic fields. The SQUID can be conceptualized in the following
way. A “beam” of electrons is split and travels along upper and a lower superconductor wires that
enclose an area in which the change in magnetic flux is to be detected. The “beams” come together again
in an area known as a weak link where an electronic interference signal is detected. Assume that the
Lagrangian describing the situation is L = Lo - (e/c)(A• v). Where A is the vector potential and v is the
velocity of the electron. Lo is the portion of the Lagrangian that is unaffected by a changing magnetic
field.
a) What is the action integral for each “path”?
b) Convert the field dependant portion of these integrals over time to path integrals.
c) Write down the transition amplitude <x’,t’|xo,to> for each “path”?
d) Write down an expression for each of the wave functions describing the upper and lower paths. You
may lump things that do not depend on the changing magnetic field together.
e) Add the two wave functions together and express the phase shift in terms of the magnetic flux through
the surface bounded by the upper and lower superconducting wires.
f) Assume that a phase shift of 2 is needed to detect a change in magnetic flux. What is the minimum
change in magnetic flux that can be detected? I am looking for a number (with units) here!
3) The Hamiltonian operator for electromagnetic radiation field can be expressed as.
 pˆ k2, 1 2 2 
ˆ
H   
  qˆk , 
2
k ,  2

a) Express this Hamiltonian in terms of the raising and lowering operators.
b) What is the ground state energy of this system? (be careful)
c) What is the energy of the first excited state?
d) What is the energy difference between the first excited state and the ground state?
The momentum operator can be expressed as
Pˆ   kaˆk† , aˆk ,
k ,
e) What is the momentum expectation value for the ground state?
f) What is the momentum expectation value for the first excited state?
g) Write down an expression which relates the answer to part d) and magnitude of part f).
4) Determine the differential scattering cross section for the potential energy V= C r -1 where C is a
constant. Assume that the incident beam energy is very large compared to the potential energy. Assume
that the incident energy is much greater than the potential.