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Problem Set 12 - Statistical Mechanics | PHYSICS 715, Assignments of Statistical mechanics

University of Wisconsin (UW) - MadisonStatistical mechanics

Material Type: Assignment; Class: Statistical Mechanics; Subject: PHYSICS; University: University of Wisconsin - Madison; Term: Spring 2006;

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PHYSICS 715
Problem Set 12 Due Friday, May 5, 2006
Reading: Landau and Lifshitz, Secs. 148-150, 152-153
Huang, Secs. 14.1-14.4, 16.1-16.4 (suggested)
FINAL EXAM MONDAY, MAY 8, 12:25 pm
LD 39: Bose-Einstein condensation in an atomic trap.
Bose-Einstein condensation was observed directly in 1995 in dilute atomic gases con-
fined in magnetic traps (see M. H. Anderson et al., Science 269, 198 (1995); C. C.
Bradley et al., Phys. Rev. Lett. 75, 1687 (1995); K. B. Davis et al., Phys. Rev. Lett.
75, 3969 (1995)). The traps are approximately harmonic with characteristic oscillation
frequencies νof about 150 Hz. The critical temperature Tcat which the condensation
starts varies with the experimental conditions, but is typically ≈150 nK.
(a) A mean field model for an ideal Bose gas in an external potential V(r) describes
the gas in a volume d3rat a point ras being in local equilibrium and having
a local chemical potential µ(r) = µ−V(r), where µis the constant chemical
potential for the entire system. When is this a reasonable approximation? The
number distribution of the particles in the volume d3ris given in this picture by a
standard Bose number distribution with µreplaced by µ(r), and the usual volume
factor replaced by d3r:
dN =1
eβ(E(p)−µ(r)−1
d3p d3r
h3=1
eβ(H(p,r))−µ−1
d3p d3r
h3.
Integrate over all positions and momenta to obtain an expression for the maximum
number of particles Nthat can be accomodated in a harmonic trap with V=
1
2mω2r2, and use the result to determine the critical temperature for Bose-Einstein
condensation in terms of Nand the parameters in H. Evaluate Tcfor N= 4×104
87Rb atoms in a 150 Hz trap (5S1/2electron and a nucleus with spin s= 3/2 in
a total angular momentum state with F= 2, mF= 2). Compare the result
to that in the first reference above. [Hint: the integral can be reduced to a
standard form by introducing scaled variables r0=r/λ, p0=λp,λ= 1/√mω,
introducing the the 6-dimensional vector x= (r0,p0), writing the volume element
as d6x, and changing to x2as the integration variable after performing the angular
integration.]
Determine the number N0of the atoms in the ground state of the system as a
function of T/Tc, and plot of the ratio N0/N versus T/Tc.
1
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PHYSICS 715

Problem Set 12 Due Friday, May 5, 2006

Reading: Landau and Lifshitz, Secs. 148-150, 152- Huang, Secs. 14.1-14.4, 16.1-16.4 (suggested)

FINAL EXAM MONDAY, MAY 8, 12:25 pm

LD 39: Bose-Einstein condensation in an atomic trap.

Bose-Einstein condensation was observed directly in 1995 in dilute atomic gases con- fined in magnetic traps (see M. H. Anderson et al., Science 269 , 198 (1995); C. C. Bradley et al., Phys. Rev. Lett. 75 , 1687 (1995); K. B. Davis et al., Phys. Rev. Lett. 75 , 3969 (1995)). The traps are approximately harmonic with characteristic oscillation frequencies ν of about 150 Hz. The critical temperature Tc at which the condensation starts varies with the experimental conditions, but is typically ≈ 150 nK.

(a) A mean field model for an ideal Bose gas in an external potential V (r) describes the gas in a volume d^3 r at a point r as being in local equilibrium and having a local chemical potential μ(r) = μ − V (r), where μ is the constant chemical potential for the entire system. When is this a reasonable approximation? The number distribution of the particles in the volume d^3 r is given in this picture by a standard Bose number distribution with μ replaced by μ(r), and the usual volume factor replaced by d^3 r:

dN =

eβ(E(p)−μ(r)^ − 1

d^3 p d^3 r h^3

eβ(H(p,r))−μ^ − 1

d^3 p d^3 r h^3

Integrate over all positions and momenta to obtain an expression for the maximum number of particles N that can be accomodated in a harmonic trap with V = 1 2 mω

(^2) r (^2) , and use the result to determine the critical temperature for Bose-Einstein condensation in terms of N and the parameters in H. Evaluate Tc for N = 4× 104 (^87) Rb atoms in a 150 Hz trap (5S 1 / 2 electron and a nucleus with spin s = 3/2 in a total angular momentum state with F = 2, mF = 2). Compare the result to that in the first reference above. [Hint: the integral can be reduced to a standard form by introducing scaled variables r′^ = r/λ, p′^ = λp, λ = 1/

mω, introducing the the 6-dimensional vector x = (r′, p′), writing the volume element as d^6 x, and changing to x^2 as the integration variable after performing the angular integration.]

Determine the number N 0 of the atoms in the ground state of the system as a function of T /Tc, and plot of the ratio N 0 /N versus T /Tc.

(b) The ground state wave function for an isotropic oscillator in three dimensions is

ψ 0 (r) =

π^3 /^4 r^30 /^2

e−r

(^2) / 2 r 02 , r 0 =

√ ¯h mω

Determine the scaled number distribution (^) N^1 d (^3) (dNr/r 0 ) of atoms in the trap and plot it as a function of r/r 0 for N 0 /N = 0, 0. 1 , 0. 3 , 1 .0 and 0 ≤ r/r 0 ≤ 10. Evaluate d^3 N/dr^3 at r = 0. Is this a high density? Explain. Compare your results qualitatively with those in the references above. [Hint: the momentum integral which appears must be evaluated numerically. Express the integrand in terms of ¯hω/kTc, T /Tc, and r/r 0 to see its structure before doing the integration. Note that for a given Tc, N = Nc.]

LD 40: The Bragg-Williams approximation for the 3-dimensional Ising model as a Landau-Ginzburg mean field theory. Mean field theory (or the Bragg-Williams approximation) gives the expression

m = μ 0 tanh

[ μ 0 H kT

Tc T

m μ 0

]

for the magnetic moment per spin in an Ising model with particles with magnetic moment μ 0. H is the applied magnetic field. The total magnetization per unit volume is M = nm, where n is the density of spins.

(a) Show that the magnetic Gibbs function has the form assumed in the Landau- Ginzburg mean field approach when calculated to order m^4 with H retained as a free variable. [Hint: rewrite the relation above as an equation for H, expand the inverse hyperbolic tangent which appears to order m^3 , and determine FM by integrating the relation H = (^) ∂M∂F. GM = F − M H. Do not eliminate H in GM .] (b) Show that Tc is in fact the critical temperature at which spontaneous magneti- zation appears at H = 0 in the mean field theory. Determine the temperature dependence of the magnetization and the magnetic susceptibility χ = ∂M/∂H for H → 0 to lowest order in |Tc − T | for T > Tc and for T < Tc, and find the critical exponents β and γ. [Hint: minimize GM at fixed H to determine M. Recall that, in the Landau approach, one is dealing with a Taylor series expansion in T as well as M , and replace T by Tc wherever it does not appear in the difference |T − Tc|. Show that your solution for M actually minimizes GM in the two temperature regions.] (c) Plot separate figures giving the approximate expressions for M/nμ 0 and χkTc/nμ^20 for 0 ≤ T /Tc ≤ 2 and values in the the ranges 0 ≤ M/nμ 0 ≤ 2 and 0 ≤ χkTc/nμ^20 ≤ 4. Include on the same figures the exact results for these quanti- ties obtained by solving the equation above and the corresponding equation for dM/dH (evaluated for H = 0). [Hint: To find the nontrivial solution for M numerically for T < Tc, find the zero of 1 − (^1) x tanh T Tc x, x = m/μ 0 .]