54 Gases: the distributions
the twoparticles canbewritten as theproduct oftwo one-particle wavefunctions,i.e.
ψ(1, 2 )=ψa( 1 )·ψb( 2 ) (5.1)where thelabelsaandbare twovalues ofthe statelabeljfor the one-particle states.
However, as a wavefunction ( 5 .1) suffers from the defect that it has no interchange
parity,itisneitherAnorS.Neverthelessit canbe combinedwithitsinterchanged
partner (an alternative solution to theSchrödinger equation withthe same properties)
to yield two wavefunctions, oneSandoneA.These are
ψS( 1 , 2 )=( 1 /√
2 )[ψa( 1 )ψb( 2 )+ψa( 2 )ψb( 1 )] (5.2)
ψψψA(1, 2 )=( 1 /√
2 )[ψa( 1 )ψb( 2 )−ψa( 2 )ψb( 1 )] (5.3)The reader may readily check that these functions possess the correct parity, namely
ψS( 1 ,2)=+ψS(2, 1 )andψψψA(1, 2)=−ψψψA( 2 ,1).The( 1 /
√
2 )factors arefor
normalization purposes only.
Equations ( 5 .2) and ( 5 .3) bring out a fundamental difference between bosons and
fermions.Consider thesituationinwhichthelabelsaandbareidentical,i.e.inwhich
the two particles are both competingfor the same state. For bosons, for which (5.2)
applies, there is no problem. (In fact ( 5 .2) gives an enhanced wavefunction compared
to (5.1).) On the other hand fermions are quite different. The wavefunction given by
(5.3) vanishes identicallywhen we seta=b.ThisisthePauli exclusion principle,
which recognizes that ‘no two identical fermions can occupy the same state’. I have
heardbosons referredto as ‘friendly particles’andfermions as ‘unfriendly’. Although
bosons enjoymultiple occupancyofstates,inafermion societyallstates are either
unoccupied or singly occupied!
Finally we may readily generalize alltheabove results to an assembly containing
Nrather than merelytwo identical particles. The interchangeargument is still valid
between any pair of particles. Actually this makes it obvious that the choice of+ 1
or−1 mustbeagenericchoice. When thefirstinterchange consideredis assigned
aparity, an inconsistencywill arise if all other interchanges are notgiven the same
parity. All electrons are fermions, and all^4 He atoms are bosons.
For our assemblyofNweaklyinteractingparticles, thegeneralizations of (5.1),
( 5 .2) and ( 5 .3) are obvious but a little tedious to write down. Equation ( 5 .1) becomes
an extendedproduct ofNone-particle terms. Forbosons, the expressionforψSis
similar to (5.2), except that it containsN!terms (everypermutation ofthe particle
co-ordinate labels)all combined witha + sign. The normalization factor becomes
( 1 /
√
N
√√
!).Similarly for fermions, the generalization of (5.3) is the one antisymmetric
arrangement oftheN!terms; thiswill have systematicallyalternatingsigns, andmay
be written neatly as a determinant in which rows give the state label and columns
the particle co-ordinate. But thevitalfeatureisthat the exclusion principle as stated
above stilloperates –ifthere are twoparticlesinthe same stateψψψAisidenticallyzero.