Identical particles –fermions and bosons 535 .2 Identical particles – fermions and bosons
The next step(stepIII) in the statistical method is to count the number of microstates
corresponding to a particulardistribution. Amicrostate ofcourse relates to a quantum
state ofallNparticles, andhere must come, therefore, thevitalrecognition that
identical gas particles are fundamentally indistinguishable from each other. They are,
therefore, effectively competingfor the same one-particle states. We can neverdefine
whichparticleisin a particular state, themicrostateis completelyspecifiedmerely
by statinghow manyparticles are in each state. This counting problem will be tackled
inthe next section,butfirst thereis anotherimportant question tobeaddressed.In
quantum mechanics there are twodifferent types ofidenticalparticle, whichobey
different rules as to how many particles can be allowed in each state. The particle
types are calledfermions andbosons.
Consider a system ofjust two identical indistinguishable particles. It is described
in quantum mechanics by a two-particle wavefunctionψ(1, 2), in which the label 1
is usedto represent allthe co-ordinates (i.e. space andspin co-ordinates) ofparticle
1 and the label 2 represents the co-ordinates of the otherparticle. Forψ(l, 2)to bea
valid wavefunction, two conditions must be satisfied. One is obviously that it should
beavalidsolution ofSchrödinger’s equation. But the secondconditionisthatit
should satisfythe correct symmetryrequirement for interchange of the two labels.
The symmetry required, that of ‘interchange parity’, can be outlined as follows.
Ifthelabelsofthe two particles areinterchanged,then anyphysicalobservable
cannot change (since the particles are identical). This implies that
ψ( 1 , 2 )=exp(iδ)ψ( 2 , 1 )sinceaphysicalobservablealwaysinvolves the productψ∗ψ.Thewavefunctions are
related bythe phase factor exp(iδ).But if we make two interchanges, then we come
to a mathematical identity, i.e.
ψ(1, 2)=exp(iδ) ψ(2, 1)=exp( 2 iδ) ψ(1, 2)Hence exp( 2 iδ)=1, andtherefore theinterchangefactor exp(iδ)must equal+ 1 or
−1, one of the two square roots of+1.
Particles which interchange co-ordinates with the+ 1 factor are called ‘bosons’,
andthosewiththe− 1 factor are ‘fermions’. We shalluse thesymbolStodescribe
the symmetric wavefunction of bosons, andAfor the antisymmetric function of
fermions. It can be shown (not here!), or it can be taken as an experimental result,
that allparticles withintegralspin (0, 1, 2, etc.) arebosons; whereas odd-half integral
spinparticles (spin^12 ,^32 ,etc.) are fermions. Therefore, for example, electrons and
(^3) He (bothhaving spin 1
2 )arefermions; photons (spin1)and
(^4) He (spin0)arebosons.
Next,let us consider the case where the particles are weakly interacting,therelevant
caseforourstatisticalphysics. ThismeansthatasolutiontoSchrödinger’sequationfor