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6: Two-State Systems
6.5: Strong force - Two state system or degenerate perturbation

6.5: Strong force - Two state system or degenerate perturbation

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The fundamental forces can be thought of as manifestations of two state systems. Consider a system comprising a proton and a neutron. The proton can decay into a neutron plus a pion, while the neutron can absorb the pion and become a proton. We can think of the system as two neutrons and a pion: the pion having two degenerate states \(|a \rangle\) and \(|b\rangle \) depending on which neutron it is located. The off-diagonal terms are now \(\langle a|\hat{V}_a|b \rangle\), where \(V_a\) is the potential energy of the pion due to neutron \(a\). The two state analysis shows that we can think of the pion hopping back and forth between the neutrons (the pion exchange mechanism). Or we can treat the system by degenerate perturbation theory and diagonalise the 2x2 matrix to find energies: \(V_{aa} \pm V_{ab}\). The ground state has a binding energy of \(|V_{ab}|\)

Note that \(V_{ab}\) involves the overlap between the state with the pion on one site and the state with the pion on the other site. Obviously this depends on the separation (R), and so there is a force between the neutrons \(dE_g/d_R\). As the nucleons move apart, the force depends on the tails of the wavefunctions, which in turn are exponentially dependent on the pion mass. Thus the strength of the strong force falls off exponentially with distance.

Note also that we have described the basis states of our two state system as ‘a proton and a neutron’, but the actual ground state is a mixture of the two. When interacting via the strong force, the nucleons lose their well-defined identity.

This picture of forces arising from exchange of ‘virtual’ particles (the pion is not observed as a free particle here) is the standard way of thinking about fundamental forces - the electromagnetic force involves ‘exchange of virtual photons’, the gravitational force ‘gravitons’ etc. These forces are long ranged (not exponentially decaying) because the particles involved have zero mass.

All of this is analogous to covalent bonding: ‘exchange of electrons’: and in each case there is still another level of understanding lurking beneath to define the potential \(V\) : QED (photons) for electron-ion bonding and QCD (gluons and quarks) for nucleon binding.