Schrödinger’s Waves: Propagation and Probability

David B. Cook

Kaisheng Li, Longji Li

The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition \[ \boxed{\mathcal R_D\!\left[(\Phi_\Psi)_*\mu_0^{\rm phys}\right] \overset{?}{=} \mu_{\rm Born,D}.} \] This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement

Can Zhang, Wei Hong Zhou, Tian CHEN

Wave propagation algorithm is a common method to study electromagnetic field propagation. Quantum wave propagation algorithm is an improved algorithm based on wave propagation algorithm to calculate the evolution of schrodinger equation in quantum mechanics. This paper mainly introduces the general idea of quantum wave propagation algorithm, and presents the general analysis process by taking one-dimensional infinite deep potential well, one-dimensional square potential barrier and time-dependent Schrodinger equation as examples. By comparing with the strict analytical solution of Schrodinger equation, the evolution image of probability wave function of micro particles with time is given, which verifies the correctness and effectiveness of the algorithm. The programming of quantum wave algorithm is not complicated, and has the characteristics of simple structure and strong practicability.

K. T. Hecht

Avi Marchewka, Z. Schuss

Discontinuous initial wave functions, or wave functions with a discontinuous derivative and a bounded support, arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a unidirectional current at the boundary of the support. We use path integrals to define current and unidirectional current, and to provide a direct derivation of the expression for current from the path-integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short-time propagation of an initial wave function with compact support for cases of both a discontinuous derivative and a discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{3/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{1/2}).$ This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{1/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{\ensuremath{-}1/2}).$ This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(\ensuremath{\Delta}t).$ This gives a decay law.

Carl S. Helrich