Chapter 2

Bosons

2.1Free Bose gas

2.1.1General properties

For a free BOSE gas the HAMILTONian is

~p

At high temperature we have classical behavior; we consider only ultra cold gases for which

where TQ is the quantum degeneracy temperature. With this HAMILTONian at T & 0 the ground state is identical to the product of the single particle ground states and the fixed average density 1 is given by

This equation defines m; two properties can be derived from it:

1. m e0 = 0 because n must be positive

1Note that we take the quantum mechanical as well as the statistical average

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Therefore, to keep the average density fixed, we have to decrease the modulus of the chemical potential with decreasing temperature:

This means, we have a macroscopic occupation of the ground state, i.e. BOSE- EINSTEIN condensation (BEC).

Further properties:

For T = 0 all particles are in the ground state with e0 = 0. But m = 0 also therefore the energy E0 of the system is independent of the number of particle and hence unphysical particle number fluctuations are expected. Further

This means that the excitations are identified with particles.

2.1.2 Superfluidity in Free Bose Gas condensate

Dissipation means creations of excitations with momentum ~p opposite to the gas velocity due to the interaction of the gas with wall. These excitations decrease the momentum of the gas and and stops it after some time. If the medium is superfluid, no such excitations occur. At T 0 no thermal excitations are present, only interactions with the walls are relevant. To find out if those excitations occur,

we have check, whether Ebefore > Eafter. To describe the situation we have to look at two different frames of reference:

Figure 2.1: Frames of reference

0 0 = ~ 0 = ~ =

Before k : E E0, P 0 (no excitations, all particles in ground state with p 0)