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Annales de la fondation Louis de Broglie/Annales de la Fondation Louis de Broglie

Derivation of the nonlinear schrodinger equation from first principles

Theodore Bodurov

Published Jan 1, 2005• 8 citations
OpenAlex

Abstract

The generic form of the nonlinear Schrodinger (NLS) equations is derived from two assumptions which are entirely inde- pendent from the postulates of quantum mechanics. This derivation is not possible if the particles are assumed to be point-like. Regard- ing the Schrodinger equation proper as a special case of the NLS equations, we argue that the results of this paper imply that the probabilistic interpretation of quantum mechanics is not necessary.

Related papers

Foundations of Quantum Mechanics: Generalizations of the Mathematical Axiomatic Derivation of the Schrödinger Equation

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An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.

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To the nonlinear quantum mechanics

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Direct derivation of a macroscopic NLS equation from the quantum theory

Hervé Leblond

The use of the matrix density formalism, and of a multiscale expansion, allows the derivation of a macroscopic nonlinear evolution equation for a short light pulse (a nonlinear Schrödinger equation), directly from the microscopic quantum mechanics equations. We consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms, to show that such a computation is possible. For their linear part, the results agree with that of the linear dispersion theory, but the obtained nonlinear coefficient differs appreciably from that derived from the computation of the so-called nonlinear susceptibilities, except in one simple particular situation.

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J. Oliveira-Cony

Abstract Schrödinger’s equation (SE) stands as one of the most recognized formulas in physics, yet its derivation and interpretation vary widely across different theoretical frameworks of quantum mechanics. In this paper, we explore five distinct approaches to obtaining SE: via the optico-mechanical analogy, a group-theoretic method, the action principle (through Hamilton–Jacobi theory and path integrals), stochastic mechanics, and direct postulation. Each derivation is analysed in light of the conceptual foundations it implies, with special attention to how it aligns with the orthodox, realistic, and statistical interpretations of quantum theory. By comparing these frameworks, it is emphasized the interpretative flexibility of SE and the philosophical implications of treating it as a derived versus a postulated equation. This comparison sheds light on the multiplicity of paths that converge on the same mathematical structure, on the particularities of these structures, and on the interpretive tensions that underlie quantum mechanics as a physical theory.

European Journal of Physics•Dec 2, 2025•0 citations

Remarks on the Schrodinger equation

Robert Carroll

Various origins of linear and nonlinear Schrodinger equations are discussed in connection with diffusion, hydrodynamics, and fractal structure. The treatment is mainly expository, emphasizing the quantum potential, with a few new observations.

arXiv:quant-ph/0401082•Jan 14, 2004•7 citations
The generic form of the nonlinear Schrodinger (NLS) equations is derived from two assumptions which are entirely inde- pendent from the postulates of quantum mechanics. This derivation is not possible if the particles are assumed to be point-like. Regard- ing the Schrodinger equation proper as a special case of the NLS equations, we argue that the results of this paper imply that the probabilistic interpretation of quantum mechanics is not necessary.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.
The use of the matrix density formalism, and of a multiscale expansion, allows the derivation of a macroscopic nonlinear evolution equation for a short light pulse (a nonlinear Schrödinger equation), directly from the microscopic quantum mechanics equations. We consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms, to show that such a computation is possible. For their linear part, the results agree with that of the linear dispersion theory, but the obtained nonlinear coefficient differs appreciably from that derived from the computation of the so-called nonlinear susceptibilities, except in one simple particular situation.
Abstract Schrödinger’s equation (SE) stands as one of the most recognized formulas in physics, yet its derivation and interpretation vary widely across different theoretical frameworks of quantum mechanics. In this paper, we explore five distinct approaches to obtaining SE: via the optico-mechanical analogy, a group-theoretic method, the action principle (through Hamilton–Jacobi theory and path integrals), stochastic mechanics, and direct postulation. Each derivation is analysed in light of the conceptual foundations it implies, with special attention to how it aligns with the orthodox, realistic, and statistical interpretations of quantum theory. By comparing these frameworks, it is emphasized the interpretative flexibility of SE and the philosophical implications of treating it as a derived versus a postulated equation. This comparison sheds light on the multiplicity of paths that converge on the same mathematical structure, on the particularities of these structures, and on the interpretive tensions that underlie quantum mechanics as a physical theory.