First-Principles Derivation of the Schrödinger Equation via Discrete Order Geometry (DOG): From Discrete Recursion to Continuous Wave Mechanics

Author: Zhang Suhang

Independent Researcher, Luoyang

Abstract

Based on the matrix evolution framework of Discrete Order Geometry (DOG), this paper strictly derives the Schrödinger equation as an effective approximation under the continuous spacetime limit starting from the discrete recursion relation \boldsymbol{v}_{k+1} = M \boldsymbol{v}_k. By introducing the minimum time step \Delta t and the Planck constant \hbar as scale conversion factors for discrete order lattices, it is proven that under the condition \Delta t \to 0, the unitarity requirement of the DOG evolution matrix M leads to its form M = \exp\left(-\frac{i}{\hbar} H \Delta t\right), from which the partial differential equation satisfied by wave functions is further deduced.

This paper clarifies the logical relationship between the continuous description of quantum mechanics and the discrete geometric essence: the Schrödinger equation is not the first principle of quantum theory, but merely a macroscopic approximation derived from discrete order evolution governed by DOG. This derivation establishes an ontological foundation of discrete geometry for quantum mechanics, and qualitatively predicts correction terms applicable at the Planck scale.

Keywords: Discrete Order Geometry; Schrödinger Equation; Discrete Recursion; Continuous Limit; Unitary Evolution; Planck Constant

1. Introduction

As the core dynamical equation of quantum mechanics, the Schrödinger equation has long been regarded as a fundamental postulate. Nevertheless, inevitable contradictions exist between its continuous and differentiable mathematical form and discrete quantum transition phenomena in the microscopic world. Discrete Order Geometry (DOG) provides a brand-new perspective: physical spacetime is essentially composed of finitely countable discrete order lattices, and all dynamical evolutions are strictly described by the discrete recursion formula \boldsymbol{v}_{k+1} = M \boldsymbol{v}_k. This paper aims to demonstrate that the Schrödinger equation arises naturally from such discrete recursion in the continuous limit, rather than serving as an independent fundamental hypothesis.

2. Review of the DOG Discrete Evolution Framework

In accordance with the algebraic foundation of DOG (refer to Discrete Order Geometry and Matrix Algebra: Essential Correspondence and Paradigm Reconstruction), the state of any isolated physical system is represented by an N-dimensional complex column vector \boldsymbol{v}_k, whose evolution satisfies:

\boldsymbol{v}_{k+1} = M \boldsymbol{v}_k, \qquad k = 0,1,2,\dots

\tag{1}

where M denotes an N\times N complex matrix termed the order evolution matrix.

Discretization of time: Let the actual physical time interval between two adjacent evolution steps be a fixed infinitesimal quantity \Delta t, namely t_k = k \Delta t. The component v_{k,i} of the state vector \boldsymbol{v}_k stands for the order amplitude of lattice \mathcal{L}_i at time t_k.

Axiomatic assumption: The order evolution of an isolated system preserves the total norm, which constitutes the geometric origin of probability conservation:

\| \boldsymbol{v}_{k+1} \|^2 = \| \boldsymbol{v}_k \|^2, \quad \forall k.

\tag{2}

This condition is equivalent to requiring the evolution matrix M to be unitary:

M^\dagger M = I.

\tag{3}

3. Mathematical Setup for the Continuous Limit

To establish the description of continuous time evolution, we define a continuous function corresponding to the discrete state vector:

\boldsymbol{\psi}(t) = \boldsymbol{v}_k \quad \text{when} \quad t = k\Delta t,

\tag{4}

which is treated as a continuously differentiable function in the limit \Delta t \to 0. The evolution equation (1) can be rewritten as:

\boldsymbol{\psi}(t+\Delta t) = M \boldsymbol{\psi}(t).

\tag{5}

Given that \Delta t is sufficiently small, the evolution matrix M must be close to the identity matrix to avoid drastic state variations. Accordingly, we perform the following expansion:

M = I - \frac{i}{\hbar} H \Delta t + O(\Delta t^2),

\tag{6}

where H represents a Hermitian matrix (operator) independent of \Delta t, and \hbar is an introduced constant with action dimension, ensuring that H possesses energy dimension. The imaginary unit i is introduced to satisfy unitarity constraints: for M to remain unitary up to the first order of \Delta t, the expansion coefficient must be an anti-Hermitian operator; hence the standard form -i/\hbar multiplied by a Hermitian operator is adopted.

Note: The Planck constant \hbar is introduced here as a scale conversion factor. Its numerical value can be determined by the minimum action principle of discrete lattices, or regarded as a fundamental constant calibrated by experimental measurements. The DOG framework does not predict its specific value, but interprets it as a conversion coefficient linking discrete spacetime step size and continuous energy scale.

4. Derivation of the Schrödinger Equation

Substitute formula (6) into formula (5):

\boldsymbol{\psi}(t+\Delta t) = \left( I - \frac{i}{\hbar} H \Delta t \right) \boldsymbol{\psi}(t) + O(\Delta t^2).

\tag{7}

Rearrange terms and divide both sides by \Delta t:

\frac{\boldsymbol{\psi}(t+\Delta t) - \boldsymbol{\psi}(t)}{\Delta t} = -\frac{i}{\hbar} H \boldsymbol{\psi}(t) + O(\Delta t).

\tag{8}

Take the limit \Delta t \to 0. The left-hand side converges to the partial time derivative \partial \boldsymbol{\psi}/\partial t, while all higher-order infinitesimal terms vanish. We thus obtain:

i\hbar \frac{\partial}{\partial t} \boldsymbol{\psi}(t) = H \boldsymbol{\psi}(t).

\tag{9}

In the position representation, each component of \boldsymbol{\psi}(t) corresponds to the discrete sampling value of the wave function at physical lattice points \psi(\mathbf{x}_i, t). When the lattice spacing satisfies \Delta x \to 0, the above equation reduces to the standard form of the Schrödinger equation:

i\hbar \frac{\partial}{\partial t} \psi(\mathbf{x}, t) = \hat{H} \psi(\mathbf{x}, t),

\tag{10}

where \hat{H} denotes the Hamiltonian operator. The full derivation of the Schrödinger equation starting from DOG discrete recursion is hereby completed.

5. Discussion

5.1 Discrete Correction Terms

If higher-order terms with respect to \Delta t are retained, the evolution equation will contain explicit discrete corrections. In particular, the exact relation M = \exp\left(-\dfrac{i H \Delta t}{\hbar}\right) renders formula (5) a rigorous solution. Under finite time step \Delta t, wave function evolution is strictly governed by exponential mapping, which corresponds to the discrete-time version of the Schrödinger equation such as the Trotter decomposition formula in quantum mechanics.

Such discrete corrections may yield observable consequences at extremely high energy scales, including:

- Minor nonlinear modifications to the energy-momentum dispersion relation;
​
- Slight breaking of Lorentz invariance originating from spacetime discreteness.

5.2 Geometric Origin of the Probabilistic Interpretation of Wave Functions

Within the DOG framework, the conservation property \| \boldsymbol{v}_k \|^2 = \sum_i |v_{k,i}|^2 originates intrinsically from the unitarity of evolution matrices. In the continuous limit, this conserved quantity evolves into the normalization condition \int |\psi|^2 d^3x = 1, namely probability conservation. Consequently, the probabilistic interpretation in quantum mechanics is not an independent postulate, but a continuous manifestation of norm conservation of discrete order quantities.

5.3 Consistency with Heisenberg Matrix Mechanics

The present derivation is firmly established on the DOG-matrix isomorphism, and forms a dual counterpart to the derivation of the Heisenberg picture. Both theoretical routes mutually confirm that the algebraic structure of matrix mechanics and the analytical formalism of wave mechanics stem from the identical discrete geometric foundation.

6. Conclusion

Starting from the fundamental DOG evolution relation \boldsymbol{v}_{k+1}=M\boldsymbol{v}_k, this paper rigorously deduces the Schrödinger equation under unitarity constraints and continuous limit assumptions. The core conclusions are summarized as follows:

1. The Schrödinger equation is not a fundamental axiom of quantum mechanics, but an effective approximation of DOG discrete recursion in the limit \Delta t \to 0.
​
2. The Planck constant \hbar emerges naturally as a conversion constant bridging discrete spacetime step scales and continuous energy scales.
​
3. Probability conservation and unitary evolution in quantum mechanics are direct corollaries of order norm conservation within the DOG system.

This derivation lays a discrete geometric ontological foundation for quantum mechanics, and predicts testable discrete deviations near the Planck scale. Follow-up research can extend this theoretical framework to relativistic quantum mechanics and quantum field theory, aiming to construct a fully discrete formulation of quantum dynamics.

References

[1] Zhang Suhang. Discrete Order Geometry (DOG) and Matrix Algebra: Essential Correspondence and Paradigm Reconstruction. 2026.
[2] Zhang Suhang. Discrete Order Geometry (DOG) and Quantum Mechanics: Ontological Unification from Discrete Lattices to Quantum Representation. 2026.
[3] Zhang Suhang. First-Principles Derivation and Geometric Interpretation of Heisenberg Matrix Mechanics Based on Discrete Order Geometry (DOG). 2026.
[4] von Neumann J. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955.
[5] Sakurai J J. Modern Quantum Mechanics. Addison-Wesley, 1994.

Appendix: Discrete-Time Schrödinger Equation

Based on the exact exponential form M = e^{-i H \Delta t / \hbar}, discrete-time evolution can be written as:

\boldsymbol{\psi}(t+\Delta t) = e^{-\frac{i H \Delta t}{\hbar}} \boldsymbol{\psi}(t).

This expression is equivalent to the exact integral form of the Schrödinger equation, and preserves all discrete spacetime effects for finite \Delta t. Discrete corrections to energy spectra for free particles can be further derived from this formula.