Is there an intuitive way to conceptualize the ground and excited states of a transmon qubit?

I understand how a transmon qubit is analogous to an LC circuit, but has an anharmonic energy spectrum due to the nonlinearity of the Josephson junction. I also understand how to write out the Hamiltonian and the significance of the contributing terms and find the energy eigenstates.

However, I am wondering if there is a good way to conceptualize what is happening in the circuit in a more physical sense, i.e., in the ground ($$|0\rangle$$) state, what are the cooper pairs doing versus what are the cooper pairs doing in the excited ($$|1\rangle$$) state? I am basically trying to understand what physically differentiates the ground and excited states.