In a previous paper, we have mathematically derived the Schrödinger equation using the construct of a Characteristic Function. We have shown that this derivation has a great number of consequences and may help to understand what Quantum Mechanics is really about. In this paper, we present another axiomatic mathematical derivation based on the construct of Boltzmann’s entropy. We also show how these two derivations are mathematically connected and obtain, from this, the positive definite phase space probability distribution function. This probability distribution function is shown to be the only one that reproduces the Schrödinger equation and maximizes entropy, while minimizing the energy. Bohmian Mechanics is considered and reinterpreted from the perspective of the mathematical results of the present approach. Some examples are worked out to give teachers interested in using this material in their classrooms some concreteness.
In a previous paper, we have mathematically derived the Schrödinger equation using the construct of a Characteristic Function. We have shown that this derivation has a great number of consequences and may help to understand what Quantum Mechanics is really about. In this paper, we present another axiomatic mathematical derivation based on the construct of Boltzmann’s entropy. We also show how these two derivations are mathematically connected and obtain, from this, the positive definite phase space probability distribution function. This probability distribution function is shown to be the only one that reproduces the Schrödinger equation and maximizes entropy, while minimizing the energy. Bohmian Mechanics is considered and reinterpreted from the perspective of the mathematical results of the present approach. Some examples are worked out to give teachers interested in using this material in their classrooms some concreteness.
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.
This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.
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.
In this paper, we present a mathematical derivation of the Schrödinger equation departing from only two axioms. We also show that, using this formal derivation process, it is possible to directly derive the Schrödinger equation in generalized curvilinear coordinate systems. This derivation is also shown to be equivalent to Feynman’s path integral approach, but goes further, allowing us to mathematically derive the Bohr-Sommerfeld quantization rules. The use of a small parameter, both in the present derivation, where it is δr, and Feynman’s derivation, where it is ϵ = δt, is also clarified in terms of the Central Limit Theorem. Therefore, the article makes a didactic transposition of the topic of quantization, allowing it to be addressed in the context of teaching Quantum Mechanics. The epistemological importance of axiomatic approaches for the mathematical derivation and the interpretation of the symbols of the theory is also considered.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics
Abstract This paper intends to provide instruction for a single lesson in an introductory course on quantum mechanics by tying together probabilistic methods from statistical mechanics, the use of complex numbers, and a more founded explanation for the derivation of Schrödinger’s time-independent equation. Here presented derivation strongly supports the Copenhagen interpretation of quantum mechanics. Niels Bohr suggested a hydrogen atom model (similar to Kepler’s sun-planet model), where the negatively-charged electron moves in orbit around the heavy positively charged nucleus, this electron having some kinetic and potential energy. Starting from Hamilton’s differential equation for Kepler like atomic model, Schrödinger introduced a new unknown real function ψ in his derivation of time-independent equation (initially, he did not recognize it as the wave function). We show that hypothetical statistical spread of action on many (indefinite) levels is mandatory for the derivation of Schrödinger’s time-independent equation. The result of this hypothesis, the statistical probability constraint formula, is used in this derivation as we look for a minimum of probability for the electron energy distribution E ψ in the whole space. The Euler–Lagrange equation, with normalization of ψ as a constraint limitation, then gives Schrödinger’s time-independent equation for real numbers. A closer inspection of the normalized function ψ shows that it could be complex and is related to the de Broglie–Bohr model of the hydrogen atom. It can be postulated that the classical solution of Schrödinger’s time-independent equation should be a normalized complex function ψ .