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I have some trouble understanding isospin symmetry.

If we consider a system of a proton and a neutron as two different iso-states of a nucleon and we wish to write the wave function, do we have to apply the symmetrization postulate and come up with an antisymmetric function (since they are fermions), or on the otherhand we aren't subject to such restriction because despite of doing some abstraction to picture neutron and proton as 2 states, they are still distinguishable particles (due to different charge and mass, for instance)?

My question is very much related to this one Why must the deuteron wavefunction be antisymmetric? but I don't fully understand the answer: are proton and neutron identical or not? Or rather, is it valid to neglect interactions to make use of the symmetrization postulate? Finally, I also believe that part of my problem understanding these ideas relies on the fact that I don't really understand what does "isospin representation" mean (see the question mentioned above), so I would appreciate some clarification on that too.

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Protons and neutrons are not identical, but they would be if electromagnetic interactions would not exists. Since you are studying the nucleus and the strong force is strong, electromagnetic interactions are just a small correction. So as a first approximation you can neglect electromagnetic corrections and thus consider them identical. Then if you want you can add later corrections due to EM effects. What you are basically doing is a series expansion in the electric charge and considering only the zeroth order.

The isospin symmetry is an approximate symmetry meaning that it is broken by a small parameter. This means that everything you compute will respect that symmetry apart from small corrections proportional to $e$ (because they must go to zero if $e=0$). Since the breaking is small, neglecting it in a first approximation still gives you a good answer.

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