Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2. The Inhomogeneous Bogoliubov Hamiltonian
is bounded [48]. When even more particles are added to the system, the only
choice to place them is to put them into the ground state. The population
of this single quantum state with a macroscopic number of particles is called
Bose-Einstein condensation.
Let us estimate the maximum number of particles in excited states. This
is reached for the maximum possible value of the chemical potential µ =
ɛ0 = 0. As we treat the ground state separately, it is adequate to compute
the maximum number of thermal particles at a given temperature in the
continuum approximation
N max
T
�
=
dɛ ρ(ɛ)
eβɛ , (2.6)
− 1
where the density of states ρ(ɛ) typically follows a power law ρ(ɛ) = Cαɛ α−1 ,
at least for the most relevant low-energy range. If the parameter α is large
enough, the integral (2.6) converges. For α > 1, the integral can be evaluated
as
N max
T = Cα(kBT ) α Γ(α)ζ(α), (2.7)
where the product of the gamma function Γ(α) = � ∞
0 dx xα−1 e −x and the
Riemann zeta function ζ(α) = � ∞
n=1 n−α is a number of order one. The
critical particle number at a given temperature is defined by Nc = N max
T (T ).
For free particles in a box with volume L d , the density of states is given
as
d Sd
ρ(ɛ) = L
2(2π) d(2m/�2 ) d
2 ɛ d
2 −1 , (2.8)
with the surface of the d-dimensional unit sphere Sd. That means, the
parameter α = d/2 depends on the dimension. True Bose-Einstein condensation
cannot occur in one or two dimensions, where the integral (2.7)
diverges. In contrast, in three dimensions the critical particle density is
nc = Nc/L3 = ζ( 3
2 ) λ−d
T , where λT =
� 2π� 2
mkBT
is the thermal de Broglie wave
length. As ζ( 3
2 ) ≈ 2.612 is of order one, this condition states that Bose-
Einstein condensation occurs, when the average particle spacing comes close
to the thermal de Broglie wave length, nc(T )λd T = O(1). Remarkably, ideal
Bose-Einstein condensation can occur at any temperature if the particle density
is high enough, or conversely, at any particle density if the temperature
is low enough.
In harmonic traps, the density of states is different from that of free-space,
namely ρ(ɛ) ∝ ɛd−1 . Consequently, Bose-Einstein condensation occurs also
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