Natural Wave-Particle Duality: How Particles and Waves Truly Coexist

In popular perceptions of quantum physics, "wave-particle duality" is often portrayed as a mysterious and counterintuitive phenomenon: an electron is "both a particle and a wave," or even "passes through two slits simultaneously." While this narrative is widespread, it obscures a more natural and coherent picture—one that requires no mysticism, no observer's magic, and no abandonment of physical reality.

From the perspective of Natural Quantum Theory (NQT), we can reason: wave-particle duality is not a fundamental law of nature, but a natural consequence of interactions between extended entities (such as electrons) and the global waves they generate under specific boundary conditions. In this image, particles and waves are not mutually exclusive "identities," but two complementary aspects of the same physical reality.

1. Particles Are Not "Points" but Structured Extended Entities

Traditional quantum mechanics often treats electrons as "point particles" with no size or internal structure. However, this assumption gives rise to a series of technical and conceptual dilemmas, including infinities, renormalization, and wave function collapse.

NQT posits that an electron is an extended charged entity with a finite scale, whose static structural dimension is defined by the Compton wavelength: λ_C = ℏ/(mₑc) (approximately 2.4×10⁻¹² meters). Though small, this scale is sufficient to support internal vibrations, rotations, and electromagnetic field distributions. For this reason, electrons can not only carry energy and momentum as "particles" but also act as "sources" to excite electromagnetic fields—i.e., waves.

This "structured" property is the foundation of wave-particle duality: a sizeless "point" could not generate extended waves; it is precisely the particle’s finite spatial distribution that makes the production and propagation of waves a natural outcome.

2. Moving Particles Excite Waves: The Natural Origin of de Broglie Wavelength

Traditional quantum theory regards the de Broglie wavelength (λ_D = h/p, where h is Planck’s constant and p is the particle’s momentum) as an "innate quantum property" of particles, but fails to explain its physical origin. In NQT, however, the de Broglie wavelength is the characteristic scale of waves excited by moving particles, whose essence can be derived through simple dynamic logic:

• A moving extended charged entity (e.g., an electron) continuously perturbs the surrounding electromagnetic field, forming "companion waves" coupled to its own motion. These waves are not independent of the particle but are a dynamic manifestation of the interaction between the particle and the field.

• The particle’s momentum p = mₑv directly determines the frequency of the waves: the greater the momentum (and thus the faster the particle moves), the higher the frequency of the electromagnetic perturbation, and the shorter the wavelength of the waves—this is the core reason why the de Broglie wavelength is inversely proportional to momentum (λ_D ∝ 1/p).

• Planck’s constant h is not a symbol of "quantum mysticism," but a quantitative measure of the coupling strength between wave energy and particle momentum, reflecting the dynamic correlation between the extended particle and the field it excites.

In short, the de Broglie wavelength is not a "suddenly appearing wave nature" of particles, but a natural product of the interaction between moving particles and the electromagnetic field—just as a ship excites water waves related to its speed, an electron in motion excites "field waves" related to its momentum, with the de Broglie wavelength as their defining scale.

3. Waves React Back on Particles: Resonance Drives State Transitions

A key insight is that waves excited by particles do not exist in isolation; instead, they feedback to modulate the particle’s own motion through "self-interaction"—a mechanism already foreshadowed in classical electrodynamics (e.g., radiation damping) but misinterpreted in traditional quantum interpretations as "virtual particle exchange" or "probabilistic transitions."

Within an atom, a bound electron does not "remain fixed in an orbit" but oscillates periodically around its equilibrium position under the Coulomb potential of the atomic nucleus. This oscillation radiates electromagnetic waves of specific frequencies, forming stable standing wave patterns—just as a guitar string can only resonate at specific pitches under fixed tension, the frequencies of these standing waves correspond to the atom’s "energy levels."

When an external wave (e.g., light) has a frequency that exactly matches the frequency of one of these standing wave patterns, resonance occurs: energy from the external wave is efficiently transferred to the electron through field coupling, increasing the amplitude of its oscillation until it breaks free from the atomic nucleus—this is the natural mechanism of the photoelectric effect. The entire process requires no "photon impact" or mysterious "quantum jump," but is merely an extension of the classical logic of wave resonance and energy transfer.

Similarly, an electron’s de Broglie waves couple to its own motion: when the propagation direction and frequency of the waves satisfy coherence conditions with the particle’s motion state, a "self-confinement" effect arises, causing the particle’s trajectory to follow the wave’s interference rules—laying the groundwork for the subsequent double-slit experiment image.

4. Global Boundary Conditions Select Wave Modes

The form and stability of waves are determined by the global environment. In atoms, molecules, or crystals, conditions such as spatial boundaries (e.g., the range of the atomic nucleus’s Coulomb potential), potential distributions, and neighboring charges act like a "sieve" to select allowed wave modes—this is the true origin of "quantization," not an artificially imposed rule.

• In a hydrogen atom, the waves excited by the electron must form closed standing waves around the atomic nucleus: only modes where the orbital circumference is an integer multiple of the wavelength can avoid self-interference cancellation and thus exist stably. This directly explains the quantization of hydrogen atom energy levels, whose essence is the natural resonance condition of waves in a finite space.

• In the double-slit experiment, when an electron (as an extended field) passes through the slits, its de Broglie waves interact with the geometric boundaries of the slits: the waves diffract and interfere behind the slits, forming a spatial intensity distribution; the detector records the localized concentration of field energy—manifesting as "particle-like" discrete clicks. In each experiment, the electron follows a definite trajectory (guided by the waves), eliminating the bizarre assumption that it "passes through two slits simultaneously" or that a "probability wave collapses."

5. The Unity of Waves and Particles: A Dynamically Coupled System

In the natural image, "particles" and "waves" are two dynamic aspects of the same physical reality, forming a closed-loop coupling through the following logic:

1. Particle (extended structure) motion → perturbs the electromagnetic field, exciting waves (with the de Broglie wavelength as the characteristic scale);

2. Wave propagation → feedback modulates the particle’s motion state through self-interaction;

3. Global boundary conditions → select stable wave modes (the root of quantization);

4. Wave resonance → triggers transitions in the particle’s energy and motion state (e.g., photoelectric effect, energy level transitions).

This is analogous to a dancer (particle) performing on a stage (space): the dancer’s movements (motion) excite air vibrations (waves), and the stage’s shape (boundary conditions) determines which tones can resonate persistently; if external music (external waves) matches the resonant frequency, the dancer will resonate and change their steps (state transition). The dancer and the sound waves are not independent entities but a dynamically coupled whole—this is the essence of the interaction between light and matter.

6. Why Does the Traditional Interpretation Seem "Bizarre"?

Mainstream quantum mechanics (instrumental quantum theory) treats the wave function ψ as a fundamental entity and the probability of measurement results as an ultimate property of nature. However, this confuses "descriptive tools" with "physical reality":

ψ is essentially a mathematical expansion coefficient of the physical field in the frequency spectrum (momentum/energy) space, much like using a Fourier series to describe a piece of music—the spectrum can accurately represent the distribution of pitches (energy/momentum), but cannot restore the physical structure of the musical instrument (the particle’s extended properties) or the propagation process of sound waves (the spatiotemporal dynamics of waves).

When we describe nature solely relying on the spectrum (e.g., energy levels, momentum distributions), we lose spatiotemporal locality, causal chains, and energy distribution—this is the root of "quantum weirdness" (mystified interpretations of nonlocality, wave function collapse, and the uncertainty principle). Once we return to the image of continuous fields and extended particles in spacetime, however, this "weirdness" naturally dissipates: all quantum phenomena are inevitable outcomes of the dynamic coupling between waves and particles, following the causal logic of classical physics.

Conclusion: Returning to an Intelligible Nature

Nature's wave-particle duality does not require us to accept the logical paradox that "an electron is both a wave and a particle," but invites us to embrace a richer physical picture: the world is composed of structured extended entities that interact through electromagnetic fields, and their behavior is jointly shaped by wave laws, momentum coupling (de Broglie wavelength), and boundary conditions.

This image needs no "God playing dice," no "observation creating reality," and no abandonment of the quest for physical mechanisms. It tells us that quantum phenomena are precisely predictable not because nature is random, but because the behavior of wave systems under resonance conditions is highly regular—the determinism of the de Broglie wavelength, the quantization of energy levels, and the repeatability of interference patterns are all direct manifestations of this regularity.

Understanding this allows us not only to grasp the essence of atoms, light, and matter more deeply but also to reclaim science’s most precious quality—faith in a knowable, intelligible natural world that exists independently of the observer.