Editing Indistinguishable particles (section)

=== Statistical properties of bosons and fermions ===

There are important differences between the statistical behavior of bosons and fermions, which are described by [[Bose–Einstein statistics]] and [[Fermi–Dirac statistics]] respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the [[laser]], [[Bose–Einstein condensate|Bose–Einstein condensation]], and [[superfluid]]ity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the [[Fermi gas]]. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within [[Electron shell|shells]] rather than all lying in the same lowest energy state.

The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled <math>|0\rangle</math> and <math>|1\rangle</math>, which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the <math>|0\rangle</math> and <math>|1\rangle</math> states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on [[quantum entanglement]].) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured.

If A and B are distinguishable particles, then the composite system has four distinct states: <math>|0\rangle|0\rangle</math>, <math>|1\rangle|1\rangle</math>, <math>|0\rangle|1\rangle</math>, and <math>|1\rangle|0\rangle</math>. The probability of obtaining two particles in the <math>|0\rangle</math> state is 0.25; the probability of obtaining two particles in the <math>|1\rangle</math> state is 0.25; and the probability of obtaining one particle in the <math>|0\rangle</math> state and the other in the <math>|1\rangle</math> state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: <math>|0\rangle|0\rangle</math>, <math>|1\rangle|1\rangle</math>, and <math>\frac{1}{\sqrt{2}}(|0\rangle|1\rangle + |1\rangle|0\rangle)</math>. When the experiment is performed, the probability of obtaining two particles in the <math>|0\rangle</math> state is now 0.33; the probability of obtaining two particles in the <math>|1\rangle</math> state is 0.33; and the probability of obtaining one particle in the <math>|0\rangle</math> state and the other in the <math>|1\rangle</math> state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump".

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state <math>\frac{1}{\sqrt{2}}(|0\rangle|1\rangle - |1\rangle|0\rangle)</math>. When the experiment is performed, one particle is always in the <math>|0\rangle</math> state and the other is in the <math>|1\rangle</math> state.

The results are summarized in Table 1:

{| class="wikitable" style="margin:auto"
|+ Table 1: Statistics of two particles
|-
! Particles !! Both 0 !! Both 1 !! One 0 and one 1
|-
| Distinguishable|| align=center |0.25|| align=center |0.25|| align=center |0.5
|-
| Bosons|| align=center |0.33|| align=center |0.33|| align=center |0.33
|-
| Fermions|| align=center |0|| align=center |0|| align=center |1
|}

As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]], these principles are extended to large number of particles, with qualitatively similar results.