Speculation on the Two-Dimensionality of the Free Particle exp(ipx)

A classical wave (string, sound, water etc) which satisfies the classical wave equation is described by a “snapshot” profile cos(-Et+px) at an instant in time. By changing t and x such that  -E dt + p dx = 0 one follows this translating profile through time and space, but physically a sound, string or water wave is not simply translating like this.The medium fills the “trough” where the wave once passed, but the medium is not part of the wave and so need not be included in its solution. There is a wave profile, but at the same time changes to the profile which are excluded.

    In this note we speculate that the form exp(-iEt+ipx) used to describe a free quantum particle (and associated with a kind of dynamic probability like the classical wave’s cos(-Et+px)) is two dimensional because it describes both a profile in space due to a kind of momentum-energy wave acting on a single particle, but also as a second dimension the magnitude of the change in this wave sin(px) unlike the classical wave approach. In other words, for a single particle which is the medium of the energy-momentum wave, it cannot be removed or excluded (hence a particle-wave duality). We suggest there should be a physical mechanism associated with this - in other words the motion of a quantum particle is not a simple translation.