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CHAPTER 12

BOSE SYSTEM

The dominant characteristic of a system of bosons is a "statistical"

attraction between the particles. In contradistinction to the case of

fermions, the particles like to have the same quantum numbers. When

the particle number is conserved, this attraction leads to the Bose-

Einstein condensation, which is the basis of superfluidity. In this chapter

we illustrate various bose systems, discuss the Bose-Einstein

condensation, and introduce the notion of the superfluid order

parameter.

12.1 PHOTONS

Consider the equilibrium properties of electromagnetic radiation

enclosed in a volume V at temperature T, a system known as a

"blackbody cavity." It can be experimentally produced by making a

cavity in any material, evacuating the cavity completely, and then

heating the material to a given temperature. The atoms in the walls of

this cavity will constantly emit and absorb electromagnetic radiation, so

that in equilibrium there will be a certain amount of electromagnetic

radiation in the cavity, and nothing else. If the cavity is sufficientlylarge, the thermodynamic properties of the radiation in the cavity should

be independent of the nature of the wall. Accordingly we can impose on

the radiation field any boundary condition that is convenient.

The Hamiltonian for a free electromagnetic field can be written as

a sum of terms, each having the form of a Hamiltonian for a harmonic

oscillator of some frequency. This corresponds to the possibility of

regarding any radiation field as a linear superposition of plane waves of

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various frequencies. In quantum theory each harmonic oscillator of

frequency can only have the energies n+12, where

n=1,2,3 .This fact leads to the concept of photons as quanta of the

electromagnetic field. A state of the free electromagnetic field is

specified by the numbern for each of the oscillators. In other words, it

is specified by enumerating the number of photons present for each

frequency.

According to the quantum theory of radiation, photons are

massless bosons of spin . The masslessness implies that a photon

always moves with the velocity of light c in' free space, and that its spin

can have only two independent orientations: parallel and antiparallel to

the momentum. A photon in a definite spin state corresponds to a plane

electromagnetic wave that is either right or left-circularly polarized. We

may, however, superimpose two photon states with definite spins and

obtain a photon state that is linearly polarized but that is not

an eigenstate of spin. In the following we consider linearly polarized

photons. For our purpose it is sufficient to know that a photon of

frequency w has the following properties :

Energy=Momentum= k, k=c

Polarization= , =1, k.=0Such a photon corresponds1 to a plane wave of electromagnetic

radiation whose electric field vector is

Er,t=eik.r-t

The direction of is the direction of the electric field. The condition

k = 0 is a consequence of the transversality of the electric field, i.e.,

1*For a precise meaning of this statement, we refer the reader to any book on thequantum theory of radiation.

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E = 0. Thus for given k there are two and only two independent

polarization vectors e. If we impose periodic boundary conditions on

E(r, t) in a cube of volume V L3, we obtain the following allowed

values of k: