[antoine-levitt]

> The name quantum in quantum theory is related to the fact that in a separable Hilbert space, any set of mutually orthonormal vectors is countable.

Pretty sure the name quantum comes from the fact that some physical phenomena (eg absorption spectrum) get discrete allowed values. That in principle has nothing to do with separability (eg you can come up with non separable spaces which have operators with discrete spectrum). In fact presenting things this way is pretty confusing since separable Hilbert spaces do support operators with continuous spectrum (which is not obvious!) As far as I know separability is mostly technical, and often added to make life a bit simpler, since it's pretty hard to come up with useful non separable Hilbert spaces.

[hovden]

A lot of non-quantum waves have discrete allowed values. EM cavities, guitar strings, etc. Quantum waves are described by a special wave equation, actually a complex diffusion equation (first order in time, second order spatially).