Identical particles 389|ΨS(ma,mb)〉 ∝|ma;mb〉+|mb;ma〉|ΨA(ma,mb)〉 ∝|ma;mb〉−|mb;ma〉. (9.4.10)Similarly, suppose we have two identical particles in one dimension, andwe perform an
experiment capable of determining the position of each particle. If the measurement
determines that one particle is at positionx=aand the other is atx=b, then
the state of the system after the measurement would be one of the two following
possibilities:
|ΨS(a,b)〉∝ |a b〉+|b a〉|ΨA(a,b)〉∝ |a b〉−|b a〉. (9.4.11)How do we know whether a given pair of identical particles will opt for the symmetric
or antisymmetric state? In order to resolve this ambiguity, the standard postulates of
quantum mechanics need to be supplemented by an additional postulate that speci-
fies which of the two possible physical states the particle pair will assume. The new
postulate states the following: In nature, particles are of two possible types – those
that arealwaysfound in symmetric (S) multiparticle states and those that arealways
found in antisymmetric (A) multiparticle states. The former are known as bosons
(named for the Indian physicist Satyendra Nath Bose (1894–1974)) and the latter as
fermions (named for the Italian physicist Enrico Fermi (1901-1954)). Fermions are
half-integer-spin particles (s= 1/ 2 , 3 / 2 , 5 /2,...), while bosons are integer-spin parti-
cles (s= 0, 1 ,2,...). Examples of fermions are electrons, protons, neutrons, and^3 He
nuclei, all of which are spin-1/2 particles. Examples of bosons are^4 He nuclei, which
are spin-0, and photons, which are spin-1. Note that the antisymmetric state has
the important property that ifma=mb,|ΨA(ma,ma)〉=|ΨA(mb,mb)〉= 0. Since
identical fermions are found in antisymmetric states, it follows thatno two identical
fermions can be found in nature in exactly the same quantum state. Put another way,
no two identical fermions can have the same set of quantum numbers. This statement
is known as thePauli exclusion principleafter the Austrian physicist Wolfgang Pauli
(1900–1958).
Suppose a system is composed ofNidentical fermions or bosons with coordinate
labelsr 1 ,...,rNand spin labelss 1 ,...,sN. The spin labels designate the eigenvalue of
Sˆzfor each particle. Let us define, for each particle, a combined labelxi≡ri,si. Then,
for a given permutationP(1),...,P(N) of the particle indices 1,..,N, the wave function
will be totally symmetric if the particles are bosons:
ΨB(x 1 ,...,xN) = ΨB(xP(1),....,xP(N)). (9.4.12)For fermions, as a result of the Pauli exclusion principle, the wave function is anti-
symmetric with respect to an exchange of any two particles in the system. Therefore,