  Exercise 4 – Boltzmann Distribution & Helmholtz Free Energy (Canonical Ensemble)

Exercise 4 – Boltzmann Distribution & Helmholtz Free Energy
(Canonical Ensemble)
1.1 Quantum harmonic oscillator
A quantum harmonic oscillator has energy levels of the form En    n  1/ 2  .
a.
b.
c.
d.
e.
Find the partition function of a single oscillator
Find the partition function of N fixed oscillators
Find the free energy
What is the average (internal) energy of the system?
What is the average (internal) energy at high temperatures? Low temperatures?
What does it mean?
1.2 The zipper problem
A zipper has N links; each link has a state in which it is closed with energy 0 and a state in
which it is open with energy  . We require, however, that the zipper can only unzip from
one side (say, the right), and that the m'th link may only open if and only if all of the links to
its right are already open. The system is kept at temperature  .
a. Find the partition function. Calculate from it the free energy and the entropy
b. Find the average number of open links at the limit of   
c. This model is a very simplified model of the unwinding of two-stranded DNA
molecules. For normal body temperature, what is the percent of open links of a DNA
coil containing approximately 3x109 links (take ε=5x10-27J)?
d. Bonus. Assume that each open link can have p states, all with energy  . Find the
partition function, and calculate the temperature in which the system changes its
properties dramatically (this phenomenon is referred to as a phase transition). What
is that change?