De Broglie’s wave hypothesis from Fisher information

Abstract

Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ . The particle, of mass m , moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density p X ( x ) , and then to an output plane of observed position y = a + z . There is no prior knowledge of the probability laws p ( a , M ) , p ( a ) or p ( M ) , with M the particle momentum at the source. It is desired to (i) optimally estimate a , in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y . Forming the ML estimate requires (iii) estimation of the likelihood law p Z ( z ) , which (iv) must obey positivity. The relation p Z ( z ) ≡ | u ( z ) | 2 ≥ 0 satisfies this. The same u ( z ) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u ( z ) achieves aims (i)–(iv). The output is parametrized by a free parameter K . For a choice K = 0 , the result is u ( z ) = δ ( z ) , indicating classical motion. Or, for a finite, empirical choice K = ħ (Planck’s constant), u ( z ) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P ( μ ) on momentum μ . For a definite momentum μ , u ( z ) becomes a sinusoid of wavelength λ = h / μ , the de Broglie result.