Do protons and neutrons have a wavefunctions?

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This might be a silly question, but do protons and neutrons have a wave function that they can be described by?
 
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Yes they do, but I don't think people ever worry about actually writing them out in nuclear physics. Protons have a wavefunction comprised of a spatial part and a spin part (and also an isospin part if you treat them as indistinguishable). For some reason nuclear physicists also assume that the neutrons and protons are bound in a more or less Coulomb potential, so that they can use the shell model to describe the nucleus. But when I took nuclear physics, we only used the nuclear wave functions at an abstract level to determine if nucleons were in a singlet or triplet state, spin-orbit coupling perturbations, etc.
 
In fact, every object in the universe has a wavefunction, I think.
 
Yes, every kind of particle has a wave function.

When we are doing atomic physics we can ignore the quantum properties of the nucleus and treat the nucleus as a stationary classical collection of particles, the reason for this is that the electron is over 1000 times less massive then the nucleons, and so it behaves more quantum mechanically.