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.
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.
We analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the context of quantum electrodynamics. Inspired by an analogy proposed by P. A. M. Dirac, Feynman showed that Schrödinger's equation could be recovered by promoting this analogy to an equality, up to a constant factor. This insight laid the foundation for the path integral formalism. In this article, we first outline the historical development of dynamical frameworks in physics and then trace Feynman's reasoning as he constructed his formulation, beginning with the double-slit experiment and its interpretation. We subsequently derive the Schrödinger equation from the path integral and demonstrate the conservation of probability. These final sections aim to provide students with a clear link between this elegant yet less commonly presented approach and the more standard methods typically taught in undergraduate courses.
Microscopic particles, like electrons, do not move following the classical laws of motion, given by Newtonian Mechanics. These particles, however, follow other laws that are associated to the propagation of waves. This becomes qualitatively clear when we observe an interference pattern arise in an experiment in which an electron beam passes through a double slit. In this brief study, we will deal with the quantum analysis of the dynamics of particles, by means of its postulates and its precise mathematical formulation. That is, we intend to introduce the wave function and the Schrodinger equation, as well as to interpret physically such a function. The scientific bases of this essay (1),(2),(3),(4),(5) can be found in the books listed in the bibliographic references.
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.
One of the important classic experiments that led to the fundamental difference between quantum and classical mechanics is the double slit experiment, which is interesting with respect to the path integral formalism since it leads to a conceptual motivation for introducing it. From Quantum Mechanics, the probability amplitude can also be determined from the Schrodinger equation as it describes the wave properties of the particles. From Quantum Mechanics, the probability amplitude (square of the probability function or wave function) can also be determined from the Schrodinger equation as it describes the wave properties of the particles. Here, the role of the trajectory of a point-like particle, formally is revived in a way compatible with the indetermination principle and is indeed the path integral approach to quantum mechanics.
In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.