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 speci…c to quantum mechanics. The only required physical assumption is conservation of momentum. 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) ju(z)j 2 0 satis…es this. The same u(z) conveniently de…nes 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) = (z), indicating classical motion. Or, for a …nite, empirical choice K =(Planck's constant), u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle's probability amplitude function P () on momentum : For a de…nite momentum ; u(z) becomes a sinusoid of wavelength = h= ; the de Broglie result.

Estimating a Repeatable Statistical Law by Requiring Its Stability During Observation

“…Principle (1) has been applied in [4][5][6][7][8][9][10] to derive wave equations of quantum mechanics, basic thermodynamics and modern cell biology. Also derived have been the phenomena of statistical physics and other sciences [6,[11][12][13][14][15][16][17][18][19]. This includes the laws of cancer growth [6,13], the near-ubiquitous occurrence of statistical power laws [19] in science, including the quarter-power laws of biological and cosmological allometry [6,19], the de Broglie wave hypothesis (now, no longer a mere hypothesis) [17], thermodynamics using Fisher information in place of entropy [6,7], the Euler equation [14] of the chemical density functional and the laws of optimum economic investment [11] and of population dynamics [6,8,18] of animate or inanimate systems.…”

“…The logic here is that the parameter a will be best estimated if, first of all, the system q(x) "behind it" is best known, as well. Nature evidently allows realization of these benefits: the EPI principle works [4,6,[11][12][13][14][15][16][17][18][19]; and the best estimate (in the sense of absolute minimum mean-square error) of a is often achievable [5,6,12,22]. This is for systems p(x) that are called "efficient", e.g., those in the exponential family.…”

Special features of the relation between Fisher information and Schrödinger eigenvalue equation

“…Fisher Information measure has been successfully applied to the study of several physical scenarios, particularly quantum mechanical ones (as a non-exhaustive recent set, see for instance [7][8][9][10][11][12][13][14][15]). We will briefly review here the pertinent formalism.…”

Interplay of information quantifiers and the modified Jaynes-Cummings model

“…and is encountered in many physical applications (see, for instance, [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], and references therein). The FIM associated to Husimi distributions Q F (X 1 X 2 ) is defined as [49] …”