It is shown that Schroedinger equation is not consistent with information theory. From the modified form of information which ensures that the most probable density function it yields tallies with a general form of continuous Riemann integrable density function that has real or imaginary zeros or singularities at end points of $[a,b] εR$, a new variational formulation for quantum mechanics is proposed that yields a system of Euler-Lagrange equations that are non-linear. It is proved that the solutions of this system are unique, orthonormal and complete. One dimensional harmonic oscillator has been solved.
It is shown that Schroedinger equation is not consistent with information theory. From the modified form of information which ensures that the most probable density function it yields tallies with a general form of continuous Riemann integrable density function that has real or imaginary zeros or singularities at end points of [a,b]εR, a new variational formulation for quantum mechanics is proposed that yields a system of Euler-Lagrange equations that are non-linear. It is proved that the solutions of this system are unique, orthonormal and complete. One dimensional harmonic oscillator has been solved.
Quantum theory is formulated as the uniquely consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if the amplitude of a quantum process can be computed in two different ways, the two answers must agree. The constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schrödinger equation must be linear: non-linear variants of quantum mechanics violate the requirement of consistency. PACS: 03.65.Bz, 03.65.Ca. The formalism of quantum mechanics is very robust; modifications are invariably unsuccessful. A number of non-linear alternative theories have been proposed, [1] experimentally tested, [2] and promptly discarded, but the question of whether new non-linear variants might prove successful is not settled. In fact, interest in the subject remains in spite of, or perhaps, partly because of
It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)
I begin by reviewing the arguments leading to a nonlinear generalisation of Schrodinger's equation within the context of the maximum uncertainty principle. Some exact and perturbative properties of that equation are then summarised: those results depend on a free regulating/interpolation parameter η. I discuss here how one may fix that parameter using energetics. Other issues discussed are, a linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation, and possible symmetry enhancements in the nonlinear theory.
In (1) the Schrodinger equation is derived using the extremization of Fisher information in a certain way formulated in (2) i.e. Extreme Physical Information. Fisher information, however, is defined in terms of P(x) which in quantum mechanics is W*(x)W(x) where W(x) is the wavefunction. As a result, the Schrodinger equation is derived in terms of P(x) and then the mathematical substitution P(x)=W(x)W(x) for a bound state “simplifies” the form of the equation. Otherwise there would be no need to introduce the wavefunction. Furthermore P(x) = constant for a quantum free particle which is also a solution of the Schrodinger equation. For P(x)=constant, however, one would normally not consider using the Fisher information. We further note that in a physical example, namely that of the reflection/refraction of light from an interface of media one is forced to introduce the concept of a wavefunction i.e. the square root of flux, otherwise the problem cannot be solved. Thus in this situation one does not solve the problem in terms of P(x) and then introduce W(x) to simplify the form. In particular, if one considers a single photon moving to the right, it either reflects or refracts (with probability), but the ensuing equations involve the weight of the square root fluxes (wavefunctions) of all three. How can this be unless there is uncertainty in time and space which allows one to have all three together? Thus the reflection/refraction problem depends on square root probability or flux and introduces the idea of uncertainty in x, t. Given the translational invariance in the problem, it is natural to introduce the translation operator d/dx. In fact, one may define the uncertainty in x such that one has an eigenfunction of -id/dx i.e. exp(ipx). This is complex and so its modulus respects P(x)=constant which would yield a Fisher’s information of 0, meaning that there is no probability and hence no need for Fisher’s information. There is, however, probability in x, but a different level than the deterministic x=vt where v=constant. Thus we argue that one may obtain exp(ipx) strictly from the notion of translational invariance. In other words, Fisher information is not required to find this form and the quantum physics associated with it. Nevertheless Fisher information and EPI (Extreme physical Information) may be used to derive the Schrodinger equation (applied to a bound state) and one may deduce exp(ipx) from this. We note, however, that Fisher information makes use of d/dx which is the translational generator and so the idea of translational invariance and Fisher’s information/EPI may be linked.
We develop a variational regularisation framework that enables analytical solutions of the stationary de~Broglie--Bohm wave equation. The formulation begins with a Fisher-information-augmented action functional for the probability density and phase fields, yielding the Madelung (Hamilton--Jacobi and continuity) equations and, upon complex recombination, a Schrödinger-type equation with a parametric information coupling μ.
Beyond this density-based formulation, we introduce a variational regularisation scheme for the de~Broglie--Bohm equations that combines a global Fisher-information regularisation at the level of the action functional with a shell-level regularisation arising from stationary flux closure. This reduction isolates the regularisation mechanism in the spatial momentum flow and yields constrained Euler--Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic structure whose admissible asymptotic branch enforces a universal canonical relation p(x)x→μ/2 near amplitude zeros.
The framework yields closed-form analytical solutions for standard potentials and reveals a systematic inverse-square regularising term in the effective potential. The associated elliptic discriminant defines a geometric length scale that, for μ=ℏ, naturally reduces to the reduced Compton wavelength. Canonical Bohmian regularisation therefore appears as a variational admissibility condition on density dynamics, producing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics.
Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction Previous paper (Causal Reconstruction: Elementary Succession and the Planck Energy) proposed that the Schrödinger equation should arise as the continuum limit of discrete causal chains. Article 18 establishes this result under explicit regularity assumptions, completing the bridge between elementary causal succession and continuous quantum dynamics. Purpose The reconstruction programme contains two distinct transitions: Discrete causal chain → continuum reconstructive action → Madelung–Schrödinger dynamics. The second step was already established in Article 5. The purpose of this paper is to prove the first step: that discrete causal chains converge to the continuous reconstructive action in a mathematically controlled way. Main Idea The reconstructive cost between two states is interpreted as a discrete Lagrangian in the sense of variational integrator theory. A causal chain therefore becomes a discrete variational system whose dynamics is governed by discrete Euler–Lagrange equations. The key observation is that the discrete action SN = Σ C(ρk,ρk+1) plays exactly the role of a discrete Lagrangian approximation of the reconstructive action Srec. Main Results Discrete Lagrangian formulation. The reconstructive cost is shown to define a discrete Lagrangian under suitable regularity conditions. Discrete Euler–Lagrange equations. Critical causal chains satisfy discrete Euler–Lagrange equations expressing momentum continuity across each causal node. Γ-convergence. The discrete action SN Γ-converges to the continuous reconstructive action Srec as the elementary action quantum tends to zero while total duration remains fixed. Convergence of solutions. Solutions of the discrete causal-chain equations converge to solutions of the continuous Madelung equations. Preservation of Kähler structure. The discrete symplectic structure converges to the canonical Kähler symplectic form of the reconstruction phase space. Recovery of Schrödinger dynamics. Combining the present result with Article 5 yields a complete chain from discrete causal succession to the Schrödinger equation. Regularity Analysis A significant contribution of the paper is the analysis of the Fisher information functional in alternative coordinates. In the density variable ρ, Fisher information becomes singular as ρ approaches zero. In the amplitude variable u = √ρ, Fisher information becomes exactly a Dirichlet energy: I(ρ) = 4||∇u||². In the wavefunction variable ψ = √ρ exp(iS/ℏ), the reconstructive action becomes smooth on the natural Hilbert space H¹(M,C). This shows that the apparent singularities of Fisher information are coordinate artefacts rather than genuine obstructions. The Three Operative Conditions The continuum limit ultimately depends on three explicit conditions: C1 — Tightness: minimising sequences remain precompact. C2 — Small-step regime: elementary causal transitions remain inside the injectivity radius. C3 — Non-degeneracy: the second variation of the reconstructive action remains positive, excluding conjugate points and ensuring local uniqueness. Condition C2 becomes automatic in the continuum limit. Conditions C1 and C3 remain the principal analytical assumptions. Conceptual Significance The paper transforms the causal-chain picture from a heuristic interpretation into a variational approximation theorem. Discrete causal succession is no longer merely compatible with quantum dynamics; it converges to it. In this view, a quantum trajectory is not a fundamental continuous object. It is the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Conclusion This paper completes the mathematical bridge between the causal layer and the quantum layer of the reconstruction programme. Under explicit regularity assumptions, discrete causal chains converge to the reconstructive action, the reconstructive action yields the Madelung equations, and the Madelung formulation is equivalent to Schrödinger dynamics. The resulting picture is a fully variational route from elementary causal succession to quantum mechanics.