In this paper, we show that the Central Limit Theorem is deeply ingrained in the mathematical and physical structure of Quantum Mechanics. We show, furthermore, that the Central Limit Theorem provides us with a clarification of the assumptions made by other quantization processes – in particular those using the notion of expansion of some variable up to second order.
In this paper, we show that the Central Limit Theorem is deeply ingrained in the mathematical and physical structure of Quantum Mechanics. We show, furthermore, that the Central Limit Theorem provides us with a clarification of the assumptions made by other quantization processes – in particular those using the notion of expansion of some variable up to second order.
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.
The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrodinger nonrelativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No prior physical laws need to be postulated. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex number system. Inferences are 1. There is a standard view that Schrodinger equation is deterministic, while wavefunction collapse is probabilistic by Born's rule. This is opposed by the now demonstrated linkage to the central limit theorem, indicating a stochastic picture for the foundation of Schrodinger equation itself. 2. This picture is also consistent with the dynamic origin of probabilities suggested for the Born rule in the de Broglie Bohm pilot wave theory. Reasons for the primary role of C are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities.
The Schrödinger equation is the fundamental equation of quantum mechanics. Since its discovery in 1926, it has been postulated — accepted as a starting point whose origin remains unexplained. This paper derives it. The derivation begins not with physics but with three conditions that any coherent structure must satisfy in order to exist at all: (i) it must be self-consistent (identity), (ii) it must be capable of persisting through change (temporality), and (iii) it must be distinguishable from its environment (distinction). These three conditions are not physical laws; they are preconditions for anything — a particle, a field, a living organism — to be a definite, persisting, identifiable entity. From these three conditions alone, a unique dynamics on the space of all logically possible configurations can be derived. This dynamics takes the form of a probability flow with exactly three components. The first is a drift that pushes the system toward self-consistent configurations — states in which no internal contradictions remain. The second is a diffusion that prevents the system from collapsing to a single rigid state, maintaining the openness and fluctuation that characterize quantum systems. The third is a circulation — a closed, non-dissipative flow that neither creates nor destroys probability but moves it in loops. This circulation is the origin of time.
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.
Purpose of the study: This study aims to provide a comprehensive pedagogical review of the Schrödinger equation by integrating its physical derivations, mathematical structure, and applications to support advanced undergraduate and beginning graduate students in understanding quantum mechanics coherently. Methodology: Literature review, pedagogical synthesis, canonical model analysis, Hilbert-space formalism, self-adjoint operator framework, spectral theory approach, quantum mechanics textbooks and journal sources, mathematical physics methods, conceptual analysis, and visualization of wave packets and quantum phenomena were used as tools and methods in this study. Main Findings: The study shows that multiple derivations of the Schrödinger equation converge to a unified structure based on linear, unitary evolution with a self-adjoint Hamiltonian. Key quantum phenomena such as superposition, tunnelling, and quantization emerge consistently from canonical models, while mathematical conditions ensure physical consistency and probability conservation. Novelty/Originality of this study: This study uniquely integrates physical derivations, rigorous mathematical structure, and pedagogical organization into a single coherent framework. It bridges conceptual gaps between theory and application, offering a unified reference that enhances understanding of quantum mechanics and supports both self-study and instructional practices.
The Schrödinger equation iℏ ∂ₜψ = Ĥψ, with non‑relativistic Hamiltonian Ĥ = ℏ²|p̂|²/(2m) + V(x̂), has been quantum mechanics’s central dynamical statement for nearly a century. It governs atomic spectra, molecular bonding, semiconductors, neutron interferometry, and countless other empirical settings. The question is why it has the form it does. In standard QM, the Schrödinger equation is a postulate. The Hilbert space is given. The momentum operator p̂ = −iℏ∇ is given. The kinetic form |p̂|²/(2m) is inherited directly from classical mechanics — quantum theory simply adopts the classical T = p²/(2m) structure without deeper justification. The first‑order time derivative is chosen because empirically it works. Each piece is accepted because the resulting equation matches experiment. This walkthrough shows how the Event Density framework derives the Schrödinger equation, in its full standard form, from a smaller set of structural commitments about what reality is made of at the substrate level. Every element — the Hilbert space, the momentum operator, linearity, first‑order time structure, the kinetic‑plus‑potential form, the factor of 1/(2m) — emerges as a forced consequence rather than a separate postulate. The derivation runs through Stone’s theorem applied twice — once on spatial translation (forcing p̂ = −iℏ∇) and once on time translation (forcing existence and uniqueness of a self‑adjoint generator with automatic linearity and first‑order structure) — closed by Galilean Lie algebra integration that identifies the time‑translation generator with the kinetic‑plus‑potential operator. The factor of 1/(2m) emerges as the chain‑rule Jacobian of the Galilean commutator [Ĥ, K̂ᵢ] = −iℏ p̂ᵢ, not as a borrowing from classical mechanics. The first‑order time derivative is a consequence of operator exponentials, not a chosen feature.