quantum mechanics - How can the nucleus of an atom be in an excited state?

An example of the nucleus of an atom being in an excited state is the Hoyle State, which was a theory devised by the Astronomer Fred Hoyle to help describe the vast quantities of carbon-12 present in the universe, which wouldn't be possible without said Hoyle State.

It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus, so one can easily see how an electron can excite from one energy level into a higher one, hence making it excited.

However, I can't see how the nucleus can get into an excited state, because clearly, they don't exist on energy levels that they can transfer between, but instead it's just a 'ball' of protons and neutrons.

So how can the nucleus of an atom be excited? What makes it excited?

Answer

First you say

It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus

By way of preparation, I'll note that in introductory course work you never attempt to handle the multi-electron atom in detail. The reason is the complexity of the problem: the inter-electron effects (screening and so on) mean that it is not simple to describe the levels of a non-hydrogen-like atom. The complex spectra of higher Z atoms attest to this.

Later you say

[nuclei] don't exist on energy levels that they can transfer between

but the best models of the nucleus that we have (shell models) do have nucleons occupying discrete orbital states in the combined field of the all the other nucleons (and the mesons that act as the carriers of the "long-range" effective strong force).

This problem is still harder than that of the non-hydrogen-like atoms because there is no heavy, highly-charged nucleus to set the basic landscape on which the players dance, but it is computationally tractable in some cases.

See my answer to "What is an intuitive picture of the motion of nucleons?" for some experimental data exhibiting (in energy space) the shell structure of the protons in the carbon nucleus. In that image you will, however, notice the very large degree of overlap between the s- and p-shell distributions. That is different than what you see in atomic orbitals because the size of the nucleons is comparable to the range of the nuclear strong force.