(564) |

(565) |

One of the fundamental postulates of quantum mechanics is the essential
*indistinguishability* of particles of the same species. What this
means, in practice, is that we cannot *label* particles of the same
species:
*i.e.*, a proton is just a proton--we cannot meaningfully talk of proton number 1 and proton number 2, *etc*.
Note that no such constraint arises in classical mechanics. Thus, in
classical mechanics particles of the
same species are regarded as being *distinguishable*,
and can, therefore, be labelled. Of course, the quantum mechanical approach is the correct one.

Suppose that we *interchange* the th and th particles:
*i.e.*,

(566) | |||

(567) |

If the particles are truly indistinguishable then nothing has changed:

Here, we have omitted the subscripts for the sake of clarity. Note that we cannot conclude that the wave-function is unaffected when the particles are swapped, because cannot be observed experimentally. Only the

where is a complex constant of modulus unity:

Suppose that we interchange the th and th particles a second time.
Swapping the th and th particles twice leaves the system completely
unchanged: *i.e.*, it is equivalent to doing nothing to the system.
Thus, the wave-functions before and after this process must be identical.
It follows from Eq. (569) that

(570) |

We conclude, from the above discussion, that the wave-function is
either completely *symmetric* under the interchange of particles, or
it is completely *anti-symmetric*. In other words,
either

In 1940 the Nobel prize winning physicist Wolfgang Pauli demonstrated,
via arguments involving relativistic invariance, that
the wave-function associated with a collection of
identical integer-spin (*i.e.*, spin , *etc*.)
particles satisfies Eq. (571), whereas the wave-function
associated with a collection of identical half-integer-spin
(*i.e.*, spin , *etc*.)
particles satisfies Eq. (572). The former type of particles
are known as *bosons* [after the Indian physicist S.N. Bose, who
first put forward Eq. (571) on empirical grounds].
The latter type of particles are called *fermions* (after the Italian
physicists Enrico Fermi, who first studied the properties of
fermion gases). Common examples of bosons are photons and atoms.
Common examples of fermions are protons, neutrons, and electrons.

Consider a gas made up of identical bosons. Equation (571) implies that the interchange of any two particles does not lead to a new state of the system. Bosons must, therefore, be considered as genuinely indistinguishable when enumerating the different possible states of the gas. Note that Eq. (571) imposes no restriction on how many particles can occupy a given single-particle quantum state .

Consider a gas made up of identical fermions. Equation (572)
implies that the interchange of any two particles does not
lead to a new physical state of the system (since
is invariant). Hence, fermions must also be
considered genuinely indistinguishable when enumerating the different possible
states of the gas. Consider the special case where particles and lie in
the same quantum state. In this case, the act of swapping the two
particles is equivalent to leaving the system unchanged, so

(574) |

Consider, for the sake of comparison, a gas made up of identical classical particles. In this case, the particles must be considered distinguishable when enumerating the different possible states of the gas. Furthermore, there are no constraints on how many particles can occupy a given quantum state.

According to the above discussion, there are *three* different sets of rules
which can be used to enumerate the states of a gas made up of identical
particles. For a boson gas, the particles must be treated as
being indistinguishable, and there is no limit to how many particles
can occupy a given quantum state. This set of rules is called *Bose-Einstein
statistics*, after S.N. Bose and A. Einstein, who first developed them.
For a fermion gas, the particles must be treated as
being indistinguishable, and there can never be more than one particle in
any given quantum state. This set of rules is called *Fermi-Dirac
statistics*, after E. Fermi and P.A.M. Dirac, who first developed them.
Finally, for a classical gas, the particles
must be treated as
being distinguishable, and there is no limit to how many particles
can occupy a given quantum state. This set of rules is called
*Maxwell-Boltzmann statistics*, after J.C. Maxwell and L. Boltzmann,
who first developed them.