In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
We develop a structural bridge between relativistic Hamilton–Jacobi theory and the relativistic Schrödinger equation within the framework of tempered distributions and Schwartz linear algebra. For translation-invariant Hamiltonians, the principal functions Sp(x)=⟨p,x⟩ restricted to the mass shell form a complete integral of the Hamilton–Jacobi equation, while their exponential images ηp=exp(ℏiSp) constitute a Schwartz basis of the tempered state space. On each spectral fiber, both classical and quantum equations reduce to the same Einstein dispersion relation. We prove that the relativistic Schrödinger equation is precisely the Schwartz–von Neumann S–linear extension of the classical energy relation from certainty momentum states to arbitrary tempered superpositions. In the presence of scalar potentials, the Hamiltonian arises as a mixed (momentum-diagonal and position-diagonal) extension, showing that the extension principle is not restricted to the free case. We further demonstrate that exact quantum dynamics cannot, in general, be represented by a single exponential phase exp(ℏiS) unless S is affine in space. Instead, quantum evolution is obtained by S–superpositions of the principal exponential family associated with a complete integral of the Hamilton–Jacobi equation. In this sense, classical elimination of parameters is replaced by linear spectral superposition. Geometrically, the exponential mapping transforms the flat affine space of Minkowski generators into a curved manifold of principal waves on which the nonlinear Hamilton–Jacobi flow pushes forward to a linear unitary Schrödinger flow. Through de Broglie–Maxwell isomorphisms, the construction extends to complex electromagnetic-like fields, preserving translation representation, dispersion relations, and polarization geometry. The results suggest that, for translation-invariant systems, quantization may be understood as an infinite-dimensional complex linearization of a classical certainty space rather than as a semiclassical approximation. Within the tempered-distribution setting, relativistic quantum dynamics emerges as the superpositional completion of a classical complete integral.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism ; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt . And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.
The Schrödinger equation is a linear partial differential equation for the wave function. Therefore, any linear combination of two solutions is also a solution. The possibility of forming linear combinations is a well-known property of vectors. Moreover, we normalize solutions for the statistical interpretation, and we have seen several examples of orthogonality relations between bound and other states. These concepts are reminiscent of the length of a vector and the angle between two vectors or the dot product. Scattering is described by a matrix, another concept from linear algebra.
In this note, we examine critically some basic features of momentum operators, both in the case of simple spatial context and in the case of spacetime. The natural time for the domain H of the momentum operator The Laurent Schwartz space Sn endowed with its own semimetric topology and with the Dirac inner product
We emphasize that the right topology to consider is the standard Schwartz topology, not the topology induced by the above Dirac inner product which needs to provide probability issues and not continuity properties. The above choice for the space H satisfy at once some fundamental requirements of the theory:0. all the functions of the space H are both smooth and square integrable;1. the operator P reveals indeed a linear endomorphism everywhere defined uponH;2. the operator is continuous with respect to the Schwartz topology;3. the operator P is uniquely extensible to the entire space of tempered distributionsSn′ and this extension is continuous with respect to the standard topology of such distribution space.4. The above unique extension shows the so-called de Broglie waves as natural eigenvectors, although those de Broglie waves are not square integrable.
The above third and fourth points induce us to consider that extension as the standard right choice for the momentum operator, not only because that extension contains in- side the previous operator defined on the Schwartz function space, but because - even more stringently - the space of temperate distributions allow us to define rigorously the fundamental concept of continuous quantum state basis and the concepts of Dirac orthogonality and Dirac normalizability for such type of bases. It happens, by the way, that some de Broglie families of quantum waves are indeed quantum basis of the space Sn′ and reveal Dirac orthogonal and Dirac normalizable.
This paper elucidates the dual structure of the Schrödinger dynamics in two correlated stages: (1) We first derive the real-valued Schrödinger equation from scratch without referring to classical mechanics, wave mechanics, nor optics, and thereby attain a concrete and clear interpretation of the Schrödinger (wave) function. Beginning with a factorization of the density distribution function of the particles to two component vectors in configuration space, we impose very simple conditions on them such as translational invariance of space-time and the conservation of flux under a given potential function. A real-valued path-integral is formulated as a Green function for the real-valued Schrödinger equation. (2) We then study a quantum stochastic path dynamics in a manner compatible with the Schrödinger equation. The relation between them is like the Langevin dynamics with the diffusion equation. Each quantum path describes a \textquotedblleft trajectory\textquotedblright\ in configuration space representing, for instance, a singly launched electron in the double-slit experiment that leaves a spot one by one at the measurement board, while accumulated spots give rise to the fringe pattern as predicted by the absolute square of the Schrödinger function. We start from the relationship between the Ito stochastic differential equation, the Feynman-Kac formula, and the associated parabolic partial differential equations, to one of which\ the Schrödinger equation is transformed. The physical significance of the quantum intrinsic stochasticity and the indirect correlation among the quantum paths and so on are discussed. The self-referential nonlinear interrelationship between the Schrödinger functions (regarded as a whole) and the quantum paths (as its parts) is identified as the ultimate mystery in quantum dynamics.