In a hydrogen atom the electron and proton spins
can combine in four ways:
the SINGLET state
has an energy
J
below the three TRIPLET states
,
and 
which are degenerate in zero magnetic field.
The energies of the singlet and triplet 
states
are not shifted much by a weak magnetic field
(where 
is the magnitude of the [negative] magnetic moment
of the triplet state),
but the
states split by
1,±1
= ±mB.
- [20 marks]
In the limit of zero magnetic field,
what fraction of an ensemble of H atoms
is in the singlet state
in thermal equilibrium at a temperature of
100 mK [0.1 K]?
ANSWER: This is a simple Boltzmann distribution: probability of singlet state
where
is the
partition function. For B=0,
, giving
.
At
J,
and
,
giving
or
= fraction of atoms in the singlet state.
- [15 marks]
Derive an expression for the polarization
of an ensemble of H atoms in a weak magnetic field
in thermal equilibrium
at a temperature
. Express your answer
in terms of
, m, B and
.
ANSWER:
or
.
- [15 marks]
Discuss the behaviour of
as
and as
.
ANSWER: As
,
and
the exponentials all approach unity; the two in
the numerator cancel, giving
.
As
,
and
. However, since
,
even faster, so
.
Of course, in between
must have
nonzero values and there must be a peak in the magnitude
somewhere. This could be found by differentiating
with respect to
and setting the derivative
equal to zero, but this was not requested.
An example is shown below for the case of
.
- [10 marks]
Show that the average number of identical fermions
in a given state of energy
in thermal and diffusive equilibrium with a reservoir
at temperature
and chemical potential
is
ANSWER: This is easy if we apply the principles of diffusive equilibrium to a given single-particle state of energy
:
it may be populated by either one particle (N=1)
or none (N=0); all other values of N are explicitly
forbidden by the Pauli principle. We use
,
where is the Gibbs sum. In this case,
.
Since
this gives
.
- [20 marks]
On one graph, sketch f as a function of
from 0 to
for two cases:
and
.
[Hint: Let
and estimate f at x=0,
and
, its minimum value.]
ANSWER: The graph below shows f as a function of
(i.e.
in units of
) for
and for
.
For z=1
(
or x=0)
we always get
and for 
(same as
or
) we always get
.
For
(same as
or
)
we get
if
and
if
. That case
() also gives
f = 0.119 at
(z=2), as shown.
The main thing to realize here is that
becomes more and more like a step function
as
gets larger and larger (i.e. at low temperature).
- [10 marks]
Describe in words what the preceding results tell us
about the occupation of various states at
.
ANSWER: At
the distribution becomes a simple step function
at
:
all states below the Fermi energy
are filled (N=1) and all states above
are empty (N=0).
- [10 marks]
Now suppose that N indentical spin-1/2 fermions
are placed in the same parabolic potential well
(simple harmonic oscillator) with fundamental frequency
.
Assume that the fermions do not interact with each other
and that 
.
What is their chemical potential
at zero temperature (also known as the
FERMI
ENERGY)?
ANSWER: Of the many ways to approach this part, the easiest is to just think about it! Since the chemical potential is the incremental change in free energy as the last particle is added (
),
and the free energy
at
,
is just the energy level at which the last particle
goes in. For spin-1/2 particles, two can go into each state
(
and
) and the states
are evenly spaced,
,
so the last particle goes in at
and that is the chemical potential at
or FERMI
ENERGY:
.
Note: it has been brought to my attention that some people had seen a derivation of the Fermi-Dirac distribution function before, while others had not; this was correlated with which Engineering specialization people had been following. Since this question is easy to answer if you grasp the basic approach and almost impossible to solve by plugging in standard formulas, it was an excellent test of your ability to synthesize ideas (the whole point of a Physics education) but an unfair one because of the importance of ``prior knowledge.'' I will try to take this into account when making up final marks, by noting which programs people are in and which courses you have taken previously. I may need to get that information from you on the Web.