Assignment 2
(1) A 1-D harmonic oscillator has an infinite series of equally spaced
energy levels E_n = n*h_cut*omega, where n is a +ve integer, h_cut = h/2*pi
and omega is the classical oscillator frequency.
(a) Obtain the free energy F. Show that at high temperatures
tau >> h_cut*omega, this becomes equal to tau*log(h_cut*omega/tau).
(b) From F obtain the entropy and specific heat capacity (at constant volume).
Make rough plots of entropy vs temperature and specific heat vs temperature.
(2) A zipper has N links say. Each link can be in one of 2 states: open (with
energy E) or closed (with energy 0). Suppose the zipper can only unzip
from one end (left, say) and the n-th link can open only if all (n-1) links to
the left of it are already open.
(a) Obtain the partition function Z.
(b) In the limit E >> temperature tau, find the average number of open links.
(3) Find the entropy of a 1-D ideal gas (i.e., ideal gas of N particles, each
of mass M having spin zero, confined to a 1-D line of length L)
at tempertaure tau.
(4) The value of the total radiant energy flux density at the Earth from Sun,
normal to incident ray, integrated over all emission wavelengths, is
the solar constant = 0.136 J/(sec cm^2).
(a) What is the total rate of energy generation of Sun (in units of J/sec) ?
(b) Given the Stefan-Boltzmann constant is 5.67 x 10^-12 J/ (sec cm^2 K^4),
show that the effective temperature of the surface of the Sun (treated as a
black body) is approximately 6000 K. The distance from Earth to Sun os
1.5 x 10^13 cm and radius of Sun is 7 x 10^10 cm.
(5) Obtain the expression for partition function of a photon gas having n modes.
From this calculate the free energy F, and integrate by parts to show that
F = -pi^2 * V * tau^4 / (45 * h_cut^3 * c^3 ).