Motivation for isospin I am having trouble understanding the point of isospin. Why do we need isospin? The rationalization that protons and neutrons are similar doesn't make sense to me.
 A: Isospin was "invented" to model experimental properties.

Isospin was introduced by Werner Heisenberg in 1932[3] to explain symmetries of the then newly discovered neutron:

*

*The mass of the neutron and the proton are almost identical: they are nearly degenerate, and both are thus often called nucleons. Although the proton has a positive charge, and the neutron is neutral, they are almost identical in all other respects.


*The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons.

For particle physics

Isospin is a dimensionless quantity associated with the fact that the strong interaction is independent of electric charge. Any two members of the proton-neutron isospin doublet experience the same strong interaction: proton-proton, proton-neutron, neutron-neutron have the same strong force attraction.

As data became accumulated other sets of particles, like the pions and the kaons showed the SU(2) symmetry of isospin, and it is a  quantum number assigned to all strongly interacting particles and resonances.
A: Quite generally in quantum mechanics, we know that whenever we have a set of states which are degeneratein energy (or mass) there is no unique way of specifying the states: any linear combination of some initially chosen set of states will do just as well, provided the normalization conditions on the states are still satisfied. As an example, consider two such states the neutron and proton. This single near coincidence of the masses was enough to suggest to Heisenberg (1932) that, as far as the strong nuclear forces were concerned (electromagnetism being negligible by comparison), the two states could be regarded as truly degenerate, so that any arbitrary linear combination of neutron and proton wavefunctions would be entirely equivalent, as far as this force was concerned, for a single ‘neutron’ or single ‘proton’ wavefunction. This hypothesis became known as the ‘charge independence of nuclear forces’. Thus redefinitions of neutron and proton wavefunctions could be allowed, of the form
\begin{eqnarray}
\psi_{p}\rightarrow \psi_{p}' &=& \alpha\psi_{p} + \beta \psi_{n}\\  
\psi_{n}\rightarrow \psi_{n}' &=& \gamma\psi_{p} + \delta \psi_{n}\\
\end{eqnarray}
for complex coefficients $\alpha$, $\beta$,$\gamma$ and $\delta$. In particular, since $\psi_{p}$ and $\psi_{n}$, we have
\begin{equation}
\hat{H}\psi_{p} = E\psi_{p}\qquad \hat{H}\psi_{n} = E\psi_{n}
\end{equation}
From which it follows that
\begin{equation}
\hat{H}\psi_{p}' = \hat{H}(\alpha\psi_{p} + \beta \psi_{n})=\alpha\hat{H}\psi_{p} + \beta \hat{H}\psi_{n}=E(\alpha\psi_{p} + \beta \psi_{n}) = E\psi_{p}'
\end{equation}
and,similarly,
\begin{equation}
\hat{H}\psi_{n}' = E\psi_{n}'
\end{equation}
showing that the redefined wave functions still describe two states with the same energy degeneracy. One can define $\psi^{(1/2)} = \binom{\psi_{p}}{\psi_{n}}$ and $V = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$. where $V$ is the indicated complex 2×2 matrix. Then we can write, $\psi^{(1/2)} \rightarrow \psi^{'(1/2)}=V\psi^{(1/2)}$.
Heisenberg’s proposal, then, was that the physics of strong interactions between nucleons remained the same under the transformation $\psi_{p/n}\rightarrow \psi_{p/n}'$: in other words, a symmetry was involved.
 From the normalization of wave functions it is obvious that $V$ has to be unitary matrix $V^{\dagger}V=1_{2\times 2}$ with determinant $\text{Det}~V = e^{i\theta}$. Where $\theta$ is a real parameter.
We can separate off such an over all phase factor from the transformations mixing‘p’and‘n’, because it corresponds to a rotation of the phase of both p and n wave functions by the same amount. Which cost to demand that $\text{Det}~V =1$.
Saying other way round, charge independence of strong nuclear force manifests itself in the form of global $SU(2)$ isospin invariance.
