de Broglie’s wave hypothesis from Fisher information

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 μμ. The particle, of mass mm, moves through free space from an unknown source-plane position aa to an unknown coordinate xx in an aperture plane of unknown probability density pX(x)pX(x), and then to an output plane of observed position y=a+zy=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 aa, 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 aa that is received at yy. Forming the ML estimate requires (iii) estimation of the likelihood law pZ(z)pZ(z), which (iv) must obey positivity. The relation pZ(z)≡|u(z)|2≥0pZ(z)≡|u(z)|2≥0 satisfies this. The same u(z)u(z) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u(z)u(z) achieves aims (i)–(iv). The output is parametrized by a free parameter KK. For a choice K=0K=0, the result is u(z)=δ(z)u(z)=δ(z), indicating classical motion. Or, for a finite, empirical choice K=ħK=ħ (Planck’s constant), u(z)u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P(μ)P(μ) on momentum μμ. For a definite momentum μ,u(z)μ,u(z) becomes a sinusoid of wavelength λ=h/μλ=h/μ, the de Broglie result.