Question

we studied the behaviour of a Bose gas in 3-dimensions and saw that, at extremely low temperatures, it gave rise to the strange phenomena of a Bose-Einstein condensate. Now, consider an ultra-cold, non-interacting Bose gas in 1-dimension.

1. (a) Write down a relation for the density of states D1(ε).

2. (b) The sum of occupancies for all the orbitals may again be written as

   N =N₀(T)+N_e(T),

   where N is the total number of atoms, N0(T) is the number of atoms in the ground state orbital, and Ne(T) is the number of atoms in all excited orbitals. Show that the integral for Ne(T) does not converge, which means that a Bose-Einstein condensate should not form in 1-D.

Answer 1

(a) The density of states (in terms of energy) for a free particle with a mass m in a d−dimensional system with a system size L is given by

,

where

.

Hence in 1D,

.

(b)

,

where

is the Bose-Einstein function with zero chemical potential.

Hence

.

By putting, , we get

.

Now, expanding

in terms of the G.P. series

,

we obtain

.

Now, by putting nx=t, we obtain

.

Now,

.

Hence

.

Now, the sum

is the well known Riemann zeta function.

Hence

.

Now, converges if the real part of s is greater than one. Hence doesn't converge. As a result, also doesn't converge.