G25.2651: Statistical Mechanics
Final Exam
Instructions: This is a take-home exam, hence lecture notes are permitted, however, books, journal articles, etc. are not! Partial credit will be given, so be sure to show ALL your work. The exam is due Monday, May 8.
, i.e., pointing along the positive z direction.
Each particle carries a charge -q (q>0), however, assume that Coulomb interactions between
the particles can be neglected. Thus, the Hamiltonian for the system is
where c is the speed of light and
is called the vector potential. 
is related to the
magnetic field 
by
One possible choice for is
Hint: Show that the Schrödinger equation 
is separable and derive the single-particle wave equation. Try a solution
to the single-particle equation of the form
and show that 
satisfies a harmonic oscillator equation
with frequency 
and equilibrium position 
.
Assume L is much larger than the range of 
.
in the high temperature ( 
) and thermodynamic
limits. Note, in this limit, it is sufficient to work to
first order in the fugacity, 
.
WARNING!! Beware of degeneracies in the energy levels besides
the spin degeneracy.
where 
is the average induced magnetization per unit volume
along the direction of the magnetic field and is given by
Calculate and 
for this system. Curie's Law for the
magnetic susceptibility states that 
. Is your
result in accordance with Curie's Law? If not, explain why it should not be.
Hint: Consider using the canonical ensemble in this case.
, where 
is the reduced mass,
about an axis through its center of
mass perpendicular to the internuclear bond axis.
The molecule is constrained to rotate in the x-y plane only.
One of the atoms carries a charge q
and the other a charge -q.
for the molecule.
.
given by
write down the perturbation Hamiltonian 
.
for 
.
Interpret your results, and in particular, explain how the allowed absorptions
and emissions are manifest in your final expression. Plot
the absorption part of your spectrum. Where do you expect the
peak intensity to occur?
Hint: Consider using a convergence factor, 
,
and let 
to go 0 at the end of the calculation.
and
m=-l,...,l is the quantum number for the z-component of angular momentum.
Where do you expect the peak intensity to
occur in the 3-dimensional case?
, 
, 
and 
for the spin-1 Ising model, for which the spin variables 
can take on the values 0, 
1.
Ising model in one dimension with h=0,
recall that the partition function could be expressed in the form
Consider a RG transformation, called the pair cell transformation,
in which 
is re-expressed as
The transfer matrix is redefined by 
. Find
the RG equation corresponding to this transformation and show that it leads
to the expected stable fixed point.
Hint: Try redefining the coupling constant by 
and show that
can be put in the same form as 
, i.e.,
and that c can be defined
implicitly in terms of u'.
Suppose that M;SPMgt;;SPMgt;m so that the motion of the y particle is very slow compared to x and that the temperature is high enough that spin statistics can be neglected. Under such conditions, it is reasonable to apply the ideas underlying the usual Born-Oppenheimer approximation to derive a path integral expression for the canonical partition function of the system. Such a situation arises in molecular systems with heavy nuclei and light electrons.
where S[x,y] is the Euclidean action functional, show that the partition function can be expressed exactly in the form
where 
is the free Euclidean action integral for the y-particle:
and where 
is the partition function of the x-particle computed along
a given path 
of the y-particle. Provide a path integral
expression for .
at a fixed value of y.
For use in the path integral, the eigenvalues 
need to
be computed along a given path, of the y-particle, which means
that the eigenvalues become functionals, 
, of this path.
Use the idea of the Born-Oppenheimer approximation to explain why
these functionals can be approximately expressed as simple imaginary time averages
of 
and provide, therefore, expressions for
and within this approximation.
for all possible values of y. Write down an approximate expression for
in this case and give a physical interpretation of your expression.