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I am fairly new to quantum physics, and I have been told that electrons, when inside an atom, only fit in discrete orbitals/wavelengths. I understand why this happens.

My question is, protons also have wave-like properties. So why aren't they constrained to the same values?

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Indeed, protons are quantum particles, they are fermions, and must also satisfy the Pauli exclusion principle (fun fact: that's why we exist). The interesting thing is that they also distribute in energy levels, that's what we call the Nuclear Shell Model.

https://en.wikipedia.org/wiki/Nuclear_shell_model

The forces are different, also the interaction, so the levels look different, but they have levels too.

I understand that this may be counterintuitive. We tend to imagine electrons as if they were in orbits (the way Bohr described them), but that is not what's going on. Electrons distribute in levels that we call orbitals, and they are levels in energy. In the space these orbitals are clouds of probability, so in reality you just know the probability of having an electron in a certain position. For this reason, we can no longer think of a planetary system like picture.

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The reason we talk of the orbitals of the electrons and not the orbitals of the proton (or the nucleus)is due to the large difference in mass. From momentum conservation if we measure a point on the electron orbital at (x,y,z,t) in hydrogen, the proton should be found if possible to measure at the same time, at a point conserving the momentum. If you look at the mass difference, that point will be the same (within a very small variation) for all points of the electron orbital.

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    $\begingroup$ The answer I was about to write. In astronomy we talk about the Earth and Moon both orbiting their barycenter, which we then approximate as the Moon orbit the Earth. The more extreme the mass ratio, the better the approximation. $\endgroup$
    – rob ♦
    May 14, 2021 at 6:14
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The hydrogen atom has discrete energy levels. These are the levels of the entire system of proton and electron. In an excited state also the proton wave function changes.

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You may want to read about magic numbers. https://en.m.wikipedia.org/wiki/Magic_number_(physics) Protons and neutrons are arranged in shells. Some arrangements are more stable than others. They call these magic numbers. From helium to unbihexium the most recognized magic numbers are 2,8, 20, 28, 50, 82 and 126

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