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2 Practice Problems on Statistical Mechanics - Problem Set 13 | PHY 831, Assignments of Statistical mechanics

Michigan State University (MSU)Statistical mechanics

Material Type: Assignment; Class: Statistical Mechanics; Subject: Physics; University: Michigan State University; Term: Fall 2002;

Typology: Assignments

Pre 2010

Uploaded on 07/23/2009

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Physics 831 - 2002
Statistical Physics
Problem Set 13
1. Show that Bose-Einstein condensation does not occur in an ideal 2D Bose gas of nonrela-
tivistic particles (6 pt)
2. Consider a system of two coupled oscillators, with coordinates q1, q2amd momenta p1, p2m
and the Hamiltonian
H=p2
1
2m+p2
2
2m+1
2X
i,j=1,2
Kijqiqj
where Kij is a positive definite 2 ×2 matrix. Show that this system can be reduced to two
independent oscillators and find their eigenfrequencies. Suppose that the system is weakly
coupled to a thermal bath. Find hq2
1i. (8pt)
Problems with numbers are from Kerson Huang, Statistical Mechanics, 2nd edition, (Wiley, NY
1987).

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Physics 831 - 2002

Statistical Physics

Problem Set 13

  1. Show that Bose-Einstein condensation does not occur in an ideal 2D Bose gas of nonrela- tivistic particles (6 pt)
  2. Consider a system of two coupled oscillators, with coordinates q 1 , q 2 amd momenta p 1 , p 2 m and the Hamiltonian H = p

(^21) 2 m +^

p^22 2 m +

∑ i,j=1, 2

Kij qiqj where Kij is a positive definite 2 × 2 matrix. Show that this system can be reduced to two independent oscillators and find their eigenfrequencies. Suppose that the system is weakly coupled to a thermal bath. Find 〈q 12 〉. (8pt)

Problems with numbers are from Kerson Huang, Statistical Mechanics, 2nd edition, (Wiley, NY 1987).