Editing Isospin (section)

== Isospin invariance ==
To a good approximation the [[proton]] and [[neutron]] have the same mass: they can be interpreted as two states of the same particle.<ref name=Greiner>{{cite book|last1=Greiner|first1=W.|last2=Müller|first2=B.|author-link1=Walter Greiner|title=Quantum Mechanics: Symmetries|year=1994|edition=2nd|isbn=978-3540580805|publisher=Springer|url=https://archive.org/details/quantummechanics0001grei|url-access=registration|page=[https://archive.org/details/quantummechanics0001grei/page/279 279]}}</ref>{{rp|141}} These states have different values for an internal isospin coordinate. The mathematical properties of this coordinate are completely analogous to [[spin (physics)|intrinsic spin]] angular momentum. The component of the operator, <math>\hat{T}_3</math>, for this coordinate has eigenvalues +{{sfrac|1|2}} and −{{sfrac|1|2}}; it is related to the charge operator, <math>\hat{Q}</math>:
<math display=block>\hat{Q} = e\left(\hat{T}_3 + \frac{1}{2}\right) </math>
which has eigenvalues <math>e</math> for the proton and zero for the neutron.<ref name=Greiner/>{{rp|144}} For a system of n nucleons, the charge operator depends upon the mass number A:
<math display=block>\hat{Q} = e\left(\hat{T}_3 + \frac{1}{2}A\right) </math>
[[Isobar (nuclide)| Isobars]], nuclei with the same mass number like <sup>40</sup>K and <sup>40</sup>Ar, only differ in the value of the <math>\hat{T}_3</math> eigenvalue. For this reason isospin is also called "isobaric spin".

The internal structure of these nucleons is governed by the [[strong interaction]], but the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the strong interaction is isospin invariant. As a consequence the nuclear forces are charge independent. Properties like the stability of [[deuterium]] can be predicted based on isospin analysis.<ref name=Greiner/>{{rp|149}} However,
this invariance is not exact and the quark model gives more precise results.