Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

2.1. Bose-Einstein condensation of the ideal gas
and annihilation operators â †
i and âi of the corresponding state. These fulfill
the bosonic commutator relations
[âi, â †
j ] = δij
[âi,
âj] = 0 = [â †
i
, ↠j ]. (2.1)
For an ideal gas of non-interacting bosons, the single-particle energy eigenstates
are occupied independently and the grand canonical partition function
factorizes into single-state partition functions
Z = �
〈{n}| e −β( ˆ H−µ ˆ � ∞�
N)
|{n}〉 = e −β(ɛi−µ)ni
� 1
= .
1 − e−β(ɛi−µ) {n}
i
ni=0
i
(2.2)
From the partition function, thermodynamic quantities like the average
energy or the average particle number can be derived. From the total number
of particles
N = kBT ∂ � 1
ln(Z) =
∂µ e
i
β(ɛi−µ)
, (2.3)
− 1
the Bose occupation number ni for the state with energy ɛi is obtained as
ni =
1
e β(ɛi−µ) − 1 . (2.4)
The chemical potential µ has to be lower than the lowest energy level, otherwise
unphysical negative occupation numbers would occur. Without loss
of generality, the lowest energy level is chosen as the origin of energy ɛ0 = 0.
All occupation numbers increase monotonically with µ, and the chemical
potential determines �
i ni = N.
2.1.2. Bose-Einstein condensation
The Bose occupation numbers ni (2.4) diverge, when the chemical potential
µ approaches the respective energy level ɛi from below. The chemical
potential has to be lower than all energy levels, so this divergence can only
happen to the occupation of the ground state. This suggests separating the
total particle number N in the ground-state population n0 and the number
of thermal particles NT
N = n0 + NT , (2.5)
with n0 = (eβ|µ| −1) −1 and NT = �
i�=0 ni. Already in 1925, Einstein pointed
out, that under certain conditions, the population of the thermal states NT
19