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‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
Thermodynamics and Statistical Mechanics I – Home Exercise 2
1.
Quantum harmonic oscillator
a. Find the entropy of a set of N oscillators of frequency  as a function of the total
( N  n  1)!
(see Kittel,
n !( N  1)!
1.55) and use the Stirling approximation for large N: ln N !  N ln N  N
b. Let U denote the total energy n  of the oscillators. Express the entropy as
N 
 (U , N ) . Show that the total energy at temperature  is U 
.
exp(  /  )  1
quantum number n. Use the multiplicity function g ( N , n) 
This is the Planck result. We will derive it later using more powerful methods that do
not require the direct calculation of the multiplicity function.
2.
Additivity of entropy for two spin systems
Given two systems of N1 ≈ N2 = 1022 spins with multiplicity functions g1(N1,s1) and g2(N2,s-s1),
the product g1g2 as a function of s1 is relatively sharply peaked at s1 = ŝ1. For s1 = ŝ1 + 1012,
the product g1g2 is reduced by 10-174 from its peak value. Use the Gaussian approximation of
the multiplicity function.
a. Compute g1g2 / (g1g2)max for s1 = ŝ1 + 1011 and s = 0
b. For s = 1020, by what factor must you multiply (g1g2)max to make it equal to
 g ( N , s ) g ( N , s  s ) ? Give the factor to the nearest order of magnitude
1
1
1
2
2
1
s1
c. How large is the fractional error in the entropy when you ignore this factor?
3.
Parametric differentiation



2
a. Evaluate the integral I  dxe x by calculating J 
2

dxe  x

0

2
 dye
 y2
(note the

borders of the integration), using polar coordinates


b. Evaluate the integral I 0  x 2 e x dx in two different ways:
2
0

Integration by parts using the known integral from (a)


Define
I ( )   e x dx and calculate it directly, then show that
2
0
I0  
I ( )
and calculate I 0
  1