If the standing wave model represents an election on the atom's orbital, what does the model represent the free election? Is the free electron experiencing any "excitement" phenomena?
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Free electrons are electrons that are not under any potential, unlike atomic electrons. Their wave functions take the form of plane waves propagating in the free space (here's a wiki page and another discussion from this site).
Do free electrons get excited? They can, but for this to happen, they need to interact with other stuff like other electrons—you need the energy from somewhere, after all. This can be modeled as the electron being in some potential. When this happens, the electron is technically not free anymore.
Mathematically, a free electron is described by the following Hamiltonian: $$ \hat{H}=\frac{\hat{p}^2}{2m} $$ while an electron under an interaction has some modification to this Hamiltonian, which makes it different from a free electron.
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