Problems and Solutions on Thermodynamics and Statistical Mechanics

234

and

ProMems d Solutions on Thermodynam'cs €4 Statistical Mechanics

(f) When T < T,, we have

2aV O3 x3f2dx

e2-1

U = F(2m)3/2(k~)5/2 1 -

312

= 0.770NkT (6).

2066

(a) In quantum statistical mechanics, define the one-particle density

(b) For a system of N identical free bosons, let

matrix in the r-representation where r is the position of the particle.

where (Nk) is the thermal averaged number of particles in the momentum
state k. Discuss the limiting behavior of pl(r) as r -+ 00, when the tem-
perature T passes from T > Tc to T < T,, where T, is the Bose-Einstein
condensation temperature. In the case lim p1 (r) approaches zero, can you

describe how it approaches zero as r becomes larger and larger?

Solution:

(a) The one-particle Hamiltonian is H = p2/2m, and the energy eigen-

states are IE). The density matrix in the energy representation is then

p(E) = exp(-E/koT), which can be transformed to the coordinate repre-

sent at ion

(rlplr') = C (rIE)(Ele-HIksTI~)(E'lr')

r+m

(SVNY, Bufldo)

E,E'

= C (PE(r)e-E'kflT6EE!pfE, (r)

E,E'

= C pE (r)e-ElkBTpfE(rP').

E