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P316
Fall 2004
Problem #1
Final
QSHO – Adsorbed Molecular Vibrations (20pts)
H2 molecules adhere to solid surfaces at low temperatures. Under some conditions,
one Hydrogen atom is bonded to the surface and the other is tethered to it by a covalent
bond and points away from the surface (parallel to x-axis). You can view this situation as a
particle of mass m (that of a Hydrogen atom) free to vibrate harmonically with angular
frequency =1013 rads-1
1. What is the ‘zero-point’ energy of each tethered H-atom?
2. Write down equations for the raising and lowering operators
â and â
(2pts)
in terms of
x̂ and p̂ , the position and momentum operators.
(5pts)
3. Write down the Hamiltonian for the vibrational states of the tethered hydrogen
atom in terms of raising and lowering operators â , â .
(4pts)
4. Write down the operator N which counts the number of energy quanta in a given
QSHO in terms of raising and lowering operators â , â .
(4pts)
5. The ground state wavefunction is of the form
 ( x)  Ce

x2
2 L2
What are the values of L and C for this problem?
(5pts)
Problem #2 Square Well (20 pts)
A
particle
in
an
infinite



square
wavefunction  ( x, t )  A 2 Sin (x / L)e
well
Et
i 1

extending
between
 Sin (2x / L)e
Et
i 2

x=0
and
x=L
has
a

 , where Ei are the energies


of stationary state i.
1.
2.
3.
4.
5.
Sketch the amplitude of this wavefunction in space at t=0.
(2pts)
Find A so that this wavefunction is normalized.
(3pts)
If a measurement of the energy is made at t=0, what are the possible results
and what is the probability of each?
(5pts)
If a measurement of the energy is made at t=T, write down equations for the
probability of detecting each of the two energies?
(5pts)
Suppose one wall of the well is still infinitely high but now has thickness a, write
an equation for the probability (as a function of a) of finding the particle
outside the well at t=0?
(5pts)
Problem #3: Superfluidity of 4He and 3He Mixtures. (20pts)
The mass of a nucleon is 6.7x10-27 Kg and 4He has one more nucleon than 3He. A
container holds a mixture of liquid 3He and liquid 4He. The total number density is
2x1028 atoms m-3 and 95% of the atoms are 4He.
What is the Bose Einstein condensation temperature TB of the 4He atoms (in
Kelvin)? This is the temperature at which the 4He atoms become a superfluid while
the 3He atoms continue to travel freely through the liquid.
(6pts)
3
2. What is the Fermi degeneracy temperature Tf of the He atoms in the 4He
superfluid (in Kelvin)?
(6pts)
3
3
3. If the He atoms pair to form bosons and then become a He superfluid at a
temperature 10-4Tf, below which temperature (in Kelvin) could you create a mixture
of a fermion and boson superfluid?
(8pts)
1.
Problem #4 MRI (20 pts)
The protons in each water molecule in your body have spin ½. When you enter a
magnetic resonance imaging (MRI) scanner, each of your protons feels a magnetic field of
B=4 Tesla along z: Assume there are ~1027 protons per cubic meter in you.
1.
What is the Larmor (angular) precession frequency for protons at this field if the
proton’s magnetic moment is
  1026 J / T
(3pts)
2. Write down the most general spinor wavefunction for the proton spin describing this

situation in terms of  the angle that  S  makes with the z-axis and B, t. (5pts)
3. If you measure the component of spin along the x-direction at time t, what is
probability as a function of time that you will get

? You can think of this as the
2
time dependence of the average x-component of spin.

M
(5pts)
4. The x-component of “magnetization vector”
of all the proton spins is
proportional to the average x-component magnetic moment
M x (t )  N / V  x (t )
where N/V is the number per cubic meter of the protons. The x-component of
magnetic field in Tesla due to this magnetization is
Bx (t )  0 M x (t ) ; 0  4  10 7 .
Now, the voltage generated in a loop of wire of area A which is being threaded by
this time dependent Bx (t ) field is given by Faraday’s Law:
V (t )   A
dBx .
dt
What is the maximum voltage generated by you in the MRI if this pickup loop has
area 10-3 m2 – hint: this occurs when =90o.
(7pts)
Problem #5 1-d Electron Gas (20 pts)
1.
In a one-dimensional electron gas show that the density of states as a function of
energy is given by:
D( E ) 
L

2m
E
where L is the length of the sample having N electrons.
2. Find an equation of the Fermi energy Ef in one–dimension
3. What is the average energy per electron at T=0K?
(6pts)
(6pts)
(8pts)