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8/2/2019 Tugas Fisika Statistik_Bab 12 Sistem Bose
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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: