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Since all classical harmonic oscillators are ellipses in phase (position-momentum) space, and since the entire phase trajectory of a given system (with a fixed rigidity and mass factor) can be specified from either the peak momentum or displacement in the trajectory, why is it said that the number of states available to the given oscillator is proportional to the area inside the phase trajectory, and not to the peak value of the position/momentum?

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I am not sure why you consider the maximum amplitude of any variable as a sensible measure of the number of states. Normally, the number of states is proportional to the energy of the system, here the area of the ellipse in phase space, so proportional to the product of the position and momentum maximum amplitudes. Absorb the units into x and p to make the ellipse into a circle.

Upon quantization, a quantum state takes up a cell of phase-space area of about ħ, (think of the uncertainty principle), and the energy is proportional to the square of the radius, so the area of that circle and, of course, to the number of states, ħn.

In classical mechanics, recall the density of states is something like dn/dE, here, constant.

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