Editing Spin–statistics theorem (section)

== Composite particles ==
The spin–statistics theorem applies not only to elementary particles but also to [[composite particles]] formed from them, provided that the internal structure of the composites is identical and they remain bound under the conditions being considered. One can consider the many-body wave function for the composite particles. If all the constituent elementary particles in one composite are simultaneously exchanged with those in another, the resulting sign change of the wave function is determined by the number of fermions within each composite.  In such systems, the total spin of the composite particle arises from the quantum mechanical [[Angular momentum coupling|addition of the angular momenta]] of its constituents: if the number of constituent fermions is even, the composite has integer spin and behaves as a boson with a symmetric wave function; if the number is odd, the spin is half-integer and the composite behaves as a fermion with an antisymmetric wave function.<ref name=":1" />  

[[Hadron]]s are composite subatomic particles made of [[quark]]s bound together by the [[strong interaction]]. Quarks are fermions with spin of 1/2. Hadrons fall into two main categories: [[baryons]], which consist of an odd number of quarks (typically three), and [[meson]]s, which consist of an even number of quarks (typically a quark and an antiquark). Baryons, such as [[proton]]s and [[neutron]]s, are fermions due to their odd number of constituent quarks. Mesons, like [[pion]]s, are bosons because they contain an even number of quarks.<ref name="Amsler-etal-2008-PDG">
{{cite journal |last=Amsler |first=C. |display-authors=etal |year=2008 |title=Quark Model |url=http://pdg.lbl.gov/2008/reviews/quarkmodrpp.pdf |journal=[[Physics Letters B]] |series=Review of Particle Physics |volume=667 |issue=1 |pages=1–6 |bibcode=2008PhLB..667....1A |doi=10.1016/j.physletb.2008.07.018 |hdl-access=free |collaboration=[[Particle Data Group]] |hdl=1854/LU-685594}}
</ref>

The effect that quantum statistics have on composite particles is evident in the superfluid properties of the two helium isotopes, [[helium-3]] and [[helium-4]]. In neutral [[atom]]s, each proton is always matched by one electron, so that the total number of protons plus electrons is always even. Therefore, an atom behaves as a fermion if it contains an odd number of neutrons, and as a boson if the number of neutrons is even. Helium-3 has one neutron and is a fermion, while helium-4 has two neutrons and is a boson. At a temperature of 2.17 K, helium-4 undergoes a phase transition to a [[superfluid]] state that can be understood as a type of Bose–Einstein condensate. Such a mechanism is not directly available for the fermionic helium-3, which remains a normal liquid to much lower temperatures. Below 2.6 mK, helium-3 also transitions into a superfluid state. This is achieved by a mechanism similar to [[superconductivity]]: the interactions between helium-3 atoms first bind the atoms into [[Cooper pairs]], which are again bosonic, and the pairs can then undergo Bose-Einstein condensation.<ref>{{Cite book |last=Leggett |first=Anthony J. |title=Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems |date=2006 |publisher=Oxford University Press |isbn=978-0-19-171195-4 |series=Oxford graduate texts|pages=3-8}}</ref><ref name=":1">{{Citation |last1=Leggett |first1=A.J. |title=Quantum Mechanics: Foundations |date=2005-01-01 |url=https://doi.org/10.1016/B0-12-369401-9/00616-1 |encyclopedia=Encyclopedia of Condensed Matter Physics |pages=99–108 |editor-last=Bassani |editor-first=Franco |access-date=2023-03-13 |place=Oxford |publisher=Elsevier |language=en |isbn=978-0-12-369401-0 |last2=Javan |first2=R. |editor2-last=Liedl |editor2-first=Gerald L. |editor3-last=Wyder |editor3-first=Peter}}</ref>

Although composite bosons exhibit similar behavior as elementary bosons, the fermionic nature of their constituents sometimes introduces subtle effects due to the [[Pauli exclusion principle]]. These effects limit how closely the composite bosons can be packed, and are especially significant in dense systems. They are sometimes modelled as effective interactions between composites.<ref>{{Cite journal |last=Combescot |first=Monique |last2=Combescot |first2=Roland |last3=Dubin |first3=François |date=2017-06-01 |title=Bose–Einstein condensation and indirect excitons: a review |url= |journal=Reports on Progress in Physics |volume=80 |issue=6 |pages=066501 |doi=10.1088/1361-6633/aa50e3 |issn=0034-4885}}</ref>