Assignment:
This exercise is for a Bose-Einstein condensate of indistinguishable atoms which do not interact with each other and are in a 3-dimensional harmonic well. The system is described by the following Hamiltonian
H = ∑ (pi²/2m + mω²xi/2) with [xj,α,pk,β] = δjkδαβ ih α, β = 1,2,3 and the sum runs from 1 to N
a) Show that the grandcanonical partition function is given by
Zgr = Π Π Π[1 − exp(−β(hω(nx + ny + nz + 3/2) − μ))]^(-1), where the products are taken over nx ,ny, nz respectively and run from 0 to ∞
The number of particles in the state with quantum numbers (nx,ny,nz) is given as
N(nx,ny,nz) = [exp(β(hω(nx + ny + nz + 3/2) − μ)) − 1]^( −1)
and the total number of particles is found to be
N = N0 + ∑ ∫dx ∫dy ∫dz exp(−lβ(hω(x + y + z + 3/2) − μ)), where N0 is the number of atoms in the Bose-Einstein condensate, the summation runs from l = 1 to ∞ and the integrals are from 0 to ∞
b) Investigate under which circumstances the ground state is occupied with macroscopically many atoms (which means N0 is very large and is proportional to the total number of particles N). Explain why μ = 3hω/2 is a suitable choice
c) Express the number of atoms in excited states for μ= 3hω/2 in terms of the Riemann-Zeta function
ζ(3) = ∑ (1/l)³.
Argue that the critical temperature of the phase transition to the Bose condensed state is given by kTc = hω ³√(N/ζ(3))