Are there individual protons and neutrons in a nucleus?
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The popular science material always talks about the number of protons and neutrons in a nucleus, but I've always wondered if that's a real thing nuclear physicists believe or if it is just a convenient model. In other words, is there some reason to believe that an atomic nucleus isn't just a single particle with mass and charge equal to a certain number of protons and neutrons instead of a clump of individual protons and neutrons? Are there experiments to distinguish one situation from the other? It's hard to imagine what such an experiment would be like. Anything that produces a fragment of a nucleus could be interpreted in either of two ways: a certain number of protons and neutrons broke off of the clump, or the single particle divided into separate pieces.
By definition, all of the electrons in an atom are indistinguishable, which can arguably be rephrased to say that any electron in the electron cloud is the same electron as any other one. When you come up with some multi-electron wavefunction that describes an atom with $n$ electrons, you technically need to add $n^2\text{-ish}$ cross terms to make sure that your wavefunction is antisymmetric under exchange of any two electrons. There's a way to do this, which I think is named for Slater, by taking the determinant of a matrix of labeled-electron states.
