Since the early 20th century, the attempt to reconcile the wave-like and particle-like behavior of the electron led to profound shifts in theoretical physics. The classical formalism of mechanics, based on Newtonian trajectories or even relativistic corrections, proved incapable of accounting for phenomena like electron diffraction, interference, and atomic energy quantization.

Following the 1924 proposal by Louis de Broglie [5] , which postulated that particles such as electrons possess a wavelength inversely proportional to their momentum (λ = h/p), experimental confirmations soon followed—most notably electron diffraction patterns that mimicked those of light waves [6] .

This profound revelation introduced the notion that matter could not be fully described by trajectories alone and that wave dynamics had to be incorporated into the theory of motion.

However, incorporating wave behavior into the classical framework proved challenging. The electromagnetic wave equations, which describe light propagation through space, could not be directly applied to matter waves like those of the electron. Electrons are localized, charged, and massive, and their wave-like behavior could not be captured by Maxwell’s equations or the classical wave equation. Attempts to use these frameworks led to contradictions: wave equations predict dispersion and infinite spreading, while electrons remain localized and discrete when measured.

The difficulty became more pronounced when trying to describe stable atomic structures. According to classical electrodynamics, an electron orbiting a nucleus would emit radiation and spiral inward due to energy loss, contradicting the observed stability of atoms. Moreover, classical mechanics could not explain the discrete energy levels of the hydrogen atom.

This prompted Erwin Schrödinger, in 1926, to formulate a new wave equation [7] —now known as the Schrödinger equation—that treats the electron not as a particle moving on a definite path, but as a distributed wavefunction ψ(x, t). The squared modulus |ψ|², interpreted by Max Born [8] , provided a probabilistic description of the likelihood of finding the electron at a given location and time. Schrödinger’s approach successfully reproduced the quantized energy levels of hydrogen and became the cornerstone of quantum mechanics.

While immensely successful in practice, the Schrödinger equation introduced a new kind of formalism: it abandoned determinism in favor of statistical interpretation, disconnected the wavefunction from physical structure, and placed measurement—and the observer—at the heart of the theory. The equation describes how the wavefunction evolves, but not what it physically represents. The electron, once seen as a definite entity with a path and a structure, became a mathematical abstraction governed by probability amplitudes.

In this work, we revisit the origin of the Schrödinger equation not to dispute its empirical validity, but to offer a new perspective grounded in physical geometry and deterministic motion. We propose that the underlying need for a probabilistic wavefunction arises from a limited view of the electron as a point-like particle. If instead we consider the electron as a self-sustained vortex—a dynamic, spatially extended structure with both internal rotation and external translation—then the wave-like properties naturally emerge from the geometry of its motion.

In this vortex-based model, the internal rotation of the electron defines a Compton-scale circulation, while its external motion corresponds to the de Broglie wavelength. The resulting trajectory is a three-dimensional helix. From this structure, quantization, interference, tunneling, and wave-like phenomena arise not from uncertainty, but from deterministic, structured motion in space-time. The need for Schrödinger’s probabilistic interpretation is thus reframed: not as a fundamental aspect of nature, but as an artifact of incomplete modeling.

Our aim is to reconstruct quantum mechanics from first principles using this vortex foundation, replacing abstract wavefunctions with real, observable, and mathematically consistent vortex motion that preserves all experimentally confirmed predictions while restoring physical intuition, determinism, and continuity to the theory.

In classical physics, wave phenomena are described by continuous, deterministic functions that evolve smoothly in space and time, representing tangible physical oscillations. For instance, sound waves in a medium like air are longitudinal mechanical waves, where the wave function describes the displacement of particles. Similarly, electromagnetic waves, such as light, are transverse oscillations of electric and magnetic fields, governed by Maxwell’s equations. These classical wave functions are well-understood and experimentally validated. However, attempts to directly apply these classical wave concepts to describe particles like electrons encountered insurmountable difficulties.

Firstly, classical waves typically propagate through a medium (e.g., air for sound) or represent disturbances in a field (e.g., the electromagnetic field for light). For the electron, no such obvious medium or underlying field was apparent. The question of what was “oscillating” to constitute the electron’s wave nature remained unanswered.

Secondly, classical waves distribute their energy continuously across the wavefront as they propagate. In contrast, electrons, despite exhibiting wave-like interference, are always detected as localized, point-like particles, depositing their entire energy at a single location. This particulate nature upon detection was incompatible with the continuous energy distribution characteristic of classical waves.

Thirdly, the phenomenon of interference, as observed in the double-slit experiment, posed a conceptual challenge. While classical waves interfere by the superposition of amplitudes from different paths, electrons were observed to build up interference patterns even when passing through the apparatus one at a time. This suggested that the electron’s wave function described a probability amplitude for a single particle rather than a classical physical wave distributed across multiple paths simultaneously.

Fourthly, the concept of wave function collapse upon measurement, a cornerstone of the Copenhagen interpretation, has no analogue in classical wave theory. Classical waves can be superposed and their components can be measured without an abrupt, discontinuous change in their state. For electrons, however, the act of measurement appeared to instantaneously collapse the wave function from a superposition of possibilities to a single, definite state. Finally, classical waves can possess a continuous spectrum of energies, typically dependent on their amplitude and frequency. Electrons in bound systems, such as atoms, however, are restricted to discrete, quantized energy levels, as evidenced by atomic spectra. This fundamental incompatibility between the continuous energy spectrum of classical waves and the quantized energy levels of electrons could not be reconciled within a purely classical framework.

In summary, the direct application of classical wave function concepts to electrons failed to account for their particulate nature upon detection, the probabilistic nature of their interference, the phenomenon of wave function collapse, and the quantization of their energy levels. This necessitated a radical departure from classical thinking, leading to the development of quantum mechanics, wherein the electron’s wave function is interpreted not as a physical wave in the classical sense, but as a mathematical construct encoding probability amplitudes.

Following de Broglie’s hypothesis that particles such as electrons exhibit wave-like properties, initial attempts were made to describe their behavior using the classical wave equation. This equation, derived from mechanical and electromagnetic wave theory, typically takes the form:

$\frac{{\partial }^2\psi }{\partial t^2}=c^2{\nabla }^2\psi$ (2.3.1)

where ψ represents a field quantity, and c is the propagation speed of the wave (e.g., the speed of light for electromagnetic fields). This second-order partial differential equation accurately describes phenomena like sound waves in air and electromagnetic waves in vacuum, assuming linear dispersion and constant propagation speed [9] .

To adapt this equation for matter waves, one might insert a plane wave solution of the form:

$\psi \left(r,t\right)=A\cos \left(k\cdot r-\omega t\right)$ (2.3.2)

where A is the amplitude, r is the position vector, t is time, k is the wave vector (|k| = 2π/λ), and ω is the angular frequency (ω = 2πf). Substituting this into the classical wave Equation (2.3.1) yields the dispersion relation:

$\omega =c\left|k\right|$ (2.3.3)

According to de Broglie, the wave properties of a particle such as an electron are connected to its momentum and energy through the relations:

$p=\hslash k$, $E=\hslash \omega$ [10] (2.3.4)

Combining these relations with the dispersion relation from the classical wave equation (ω = c|k|) leads to:

$E=pc$ (2.3.5)

This fundamental discrepancy demonstrates that the classical wave equation, in its standard form, cannot adequately describe the wave properties of massive particles like electrons. It fails to incorporate the correct energy-momentum relationship and thus cannot account for their dynamics. This failure underscored the necessity for a new theoretical framework, which eventually emerged in the form of Schrödinger’s wave mechanics and, later, relativistic quantum mechanics.

The Schrödinger equation stands as a seminal achievement in 20th-century theoretical physics. Formulated by Erwin Schrödinger in 1925-1926, it provides the fundamental mathematical framework for describing the behavior of quantum systems [11] .

The time-dependent Schrödinger equation governs the evolution of a quantum state $\Psi \left(r,t\right)$ over time:

$i\hslash \frac{\partial \Psi \left(r,t\right)}{\partial t}=\hat{H}\Psi \left(r,t\right)$ (2.4.1)

(a complex-valued function of position r and time t), ħ is the reduced Planck constant (h/2π), i is the imaginary unit, and $\hat{H}$ is the Hamiltonian operator, representing the total energy of the system. For a single non-relativistic particle of mass m moving in a potential V(r), the Hamiltonian is:

$\hat{H}=-\frac{{\hslash }^2}{2m}{\nabla }^2+V\left(r\right)$ (2.4.2)

If the Hamiltonian is time-independent, the wave function can be separated as $\Psi \left(r,t\right)=\psi \left(r\right){\text{e}}^{-iEt/\hslash }$, leading to the time-independent Schrödinger equation:

$\hat{H}\psi \left(r\right)=E\psi \left(r\right)$ (2.4.3)

Or, more explicitly:

$\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V\left(r\right)\right]\psi \left(r\right)=E\psi \left(r\right)$ (2.4.4)

where E represents the energy of the system, and ψ(r) are the corresponding energy eigenstates or stationary states.

The wave function Ψ itself is not directly observable. According to the Born rule [12] , ${\left|\Psi \left(r,t\right)\right|}^2$ represents the probability density of finding the particle at position r at time t. This probabilistic interpretation signifies a fundamental departure from classical determinism. The wave function must be continuous, normalizable ( ${\displaystyle \int {\left|\Psi \right|}^2\text{d}V}=1$), and satisfy system-specific boundary conditions.

If the potential V(r) is not time-dependent, the wave function can be separated into spatial and temporal components:

$\psi \left(r,t\right)=\psi \left(r\right){\text{e}}^{-iEt/\hslash }$ (2.4.1.1)

Substituting into the time-dependent equation yields the time-independent Schrödinger equation:

$\hat{H}\psi \left(r\right)=E\psi \left(r\right)$ (2.4.1.2)

or explicitly:

$\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V\left(r\right)\right]\psi \left(r\right)=E\psi \left(r\right)$ (2.4.1.3)

here, E represents the energy eigenvalues, and ψ(r) are the corresponding stationary states.

Although the Schrödinger equation yields precise predictions for quantum systems, its interpretation poses significant philosophical challenges. The wave function $\Psi \left(r,t\right)$ is not directly observable; its modulus squared ∣Ψ∣² gives the probability density for locating a particle in space and time, according to the Born rule.

Mathematically, the Schrödinger equation exhibits two key properties:

linearity (allowing for superposition of states) and unitarity (preserving total probability over time).

A significant interpretational challenge arises from the measurement problem: while the Schrödinger equation describes a deterministic evolution of Ψ, measurement outcomes are probabilistic. This leads to the concept of wave function collapse, where, upon measurement, Ψ instantaneously transitions to a specific eigenstate—a process not described by the Schrödinger equation itself.

Furthermore, Heisenberg’s uncertainty principle imposes fundamental limits on the simultaneous precision with which complementary variables (e.g., position and momentum, ΔxΔp ≥ ħ/2) can be known. This inherent uncertainty reinforces the probabilistic nature of quantum mechanics. The Copenhagen interpretation, historically dominant, accepts these features as intrinsic aspects of nature, viewing the wave function as representing knowledge about the system rather than its objective physical reality.

These interpretational difficulties have motivated the search for alternative formulations, including deterministic approaches.

Among the various proposals aimed at restoring determinism to quantum physics, the pilot-wave theory, also known as Bohmian mechanics or the de Broglie-Bohm theory, stands as one of the most developed and conceptually distinct alternatives. First proposed by Louis de Broglie in 1927 [21] and later independently rediscovered and extended by David Bohm in 1952 [22] , this theory posits that quantum particles possess definite positions at all times and follow deterministic trajectories. These trajectories are guided by a wave function, which itself evolves according to the standard Schrödinger equation.

In this framework, both the particle and the wave are considered ontologically real. The wave function does not merely describe the probability of finding a particle but actively influences its motion through a guiding equation. For a non-relativistic particle, the velocity is determined by the gradient of the phase of the wave function. A key feature of Bohmian mechanics is the emergence of a “quantum potential”, an additional term in the equations of motion that depends on the curvature of the amplitude of the wave function. This quantum potential is responsible for characteristically quantum phenomena, such as interference in the double-slit experiment and quantum tunneling. It is non-local, meaning it can depend on the configuration of the wave function across all space, thus naturally accommodating quantum entanglement without invoking wave function collapse. Bohmian mechanics is empirically equivalent to standard quantum mechanics; it reproduces all of its statistical predictions, provided that the initial distribution of particle positions is assumed to conform to the Born rule (i.e., $\rho \left(x,0\right)={\left|\Psi \left(x,0\right)\right|}^2$), an assumption known as the “quantum equilibrium hypothesis”. While it resolves the measurement problem by denying that measurements are fundamentally different from other physical processes (there is no collapse), it introduces its own set of conceptual challenges, including the nature of the wave function in a multi-particle system (which resides in configuration space) and the implications of its inherent non-locality in a relativistic context. Despite these challenges, pilot-wave theory demonstrates that a deterministic underpinning for quantum phenomena is mathematically consistent and empirically viable, thereby challenging the notion that indeterminism is an unavoidable feature of the quantum world. It serves as a prominent example of efforts to provide a more complete and potentially more intuitive understanding of quantum reality.

While both the vortex-based model and Bohmian mechanics seek to restore determinism and realism to quantum theory, they differ fundamentally in ontology, mathematical formalism, and physical interpretation.

In Bohmian mechanics, particles are point-like objects guided by a pilot wave (the wave function), which evolves according to the Schrödinger equation. The wave function exists in a high-dimensional configuration space and exerts a nonlocal influence on particle trajectories via the quantum potential.

In contrast, the vortex model proposes that the electron is not a point particle but a self-sustaining, irrotational vortex in a structured, superfluid-like vacuum. Its internal geometry—including circulation, rotational velocity, and phase structure—defines its quantum properties. The wave function in this model is not an abstract guiding field but a direct expression of the electron’s real, three-dimensional helical motion in physical space.

Bohmian mechanics retains the standard Schrödinger wave function and interprets it as a real entity that guides particle trajectories through a quantum potential. This potential introduces nonlocal interactions and is responsible for interference and entanglement effects.

The vortex model, by contrast, derives a new wave function from first principles of vortex motion. This function incorporates both translational (de Broglie-scale) and rotational (Compton-scale) phase components, resulting in a modified Schrödinger-like equation. Interference, spin, and tunneling emerge not from an external guiding wave but from the geometry and dynamics of the vortex itself.

In Bohmian mechanics, the quantum potential plays a central and somewhat abstract role. It governs the acceleration of particles based on the curvature of the wave function and is responsible for the non-classical aspects of motion.

The vortex model eliminates the need for a quantum potential. Instead, the internal rotational dynamics of the vortex and its interaction with vacuum elasticity generate the observed quantum behaviors. Quantization arises naturally from geometric constraints and boundary conditions of vortex stability.

Both models accommodate quantum nonlocality, but through different mechanisms. Bohmian mechanics invokes instantaneous action-at-a-distance through the configuration space wave function. This approach, while mathematically consistent, raises tension with relativistic causality.

In the vortex model, nonlocal correlations arise from phase coherence and conservation of angular momentum across spatially separated but dynamically entangled vortex systems. The vacuum medium acts as a continuous field supporting instantaneous phase alignment without requiring superluminal signaling, thus offering a more physically intuitive account of entanglement.

Bohmian mechanics assumes the Born rule as a postulate via the quantum equilibrium hypothesis. In contrast, the vortex model offers a pathway to derive the Born rule from deterministic principles. Statistical distributions of detection events arise from ensemble phase averaging, chaotic core oscillations, and ergodic internal dynamics—making the Born rule an emergent, not axiomatic, property (see Table 1 ).

In conclusion, the vortex model complements and advances the deterministic agenda initiated by Bohmian mechanics. While Bohm reintroduced causality and realism, the vortex framework grounds these principles in a physically visualizable and mathematically unified theory that may bridge quantum mechanics, classical fields, and relativity in a single coherent model.

In this model, the electron comprises two essential motions:

1) Internal rotation at the speed of light along a circular path with a radius equal to the reduced Compton wavelength, (ƛ_C = ħ/mc, where ħ is the reduced Planck constant and m is the electron mass).

2) External translational motion with velocity, producing a de Broglie wavelength, (λ_dB = h/p, where p is the electron’s momentum).

These motions generate a helical trajectory in spacetime, unifying the wave-particle duality into a single geometric configuration. The vortex is considered irrotational at the core, with streamlines forming concentric spirals. The superfluid medium ensures frictionless circulation, stabilizing the vortex through vacuum elasticity and conserving angular momentum.

By applying classical hydrodynamics to the vortex in a compressible, elastic vacuum, key physical quantities are derived:

This model reproduces observed quantities such as the Compton wavelength, de Broglie wavelength, Planck constant, and even predicts the electron’s minimum time cycle and density. It proposes that the vortex core acts as a site of vacuum breakdown, forming a field-less void where centrifugal and centripetal forces balance.

A longstanding mystery in quantum physics is the origin and meaning of electron spin. While standard quantum mechanics attributes spin to an intrinsic angular momentum without spatial rotation, the vortex model offers a classical, geometric explanation. In this framework, spin arises naturally from the internal rotational dynamics of the vortex.

This interpretation was elaborated in the author’s previous work, “A New Theory for the Essence and Origin of Electron Spin” [4] , where spin is shown to emerge from the differential rotational dynamics between the vortex core and its boundaries. The electron is modeled as a frictionless irrotational vortex in a superfluid vacuum, whose angular momentum is conserved and equivalent to Planck’s constant.

The electron vortex rotates internally at the speed of light, and this motion is constrained by the reduced Compton wavelength ${\lambda }_c=\frac{\hslash }{mc}$ giving a vortex radius $r=\frac{{\lambda }_c}{2\pi }=\frac{\hslash }{mc}$.

The angular momentum (spin) then becomes:

$L_z=I{\omega }_{rot}=\left(m{r}^2\right)\left(c/r\right)=mcr=mc\left(\hslash /mc\right)=\hslash$

This total angular momentum matches the quantum prediction. However, quantum mechanics measures only the projection of spin along an axis, yielding:

$S_z=\pm \hslash /2$

The vortex model explains this discrepancy. During the formation of the vortex, a differential rotation arises between the vortex core and its boundaries. The vortex core completes two full rotations (2 × 360˚) for every single rotation (360˚) at the boundary. This ratio manifests as the spin-1/2 behavior: the vortex must rotate 720 degrees to return to its original configuration.

Thus, the spin quantum number 1/2 reflects this intrinsic geometric lag in vortex rotation—a physical, measurable consequence of substructure rather than an abstract property. The quantization of spin follows from boundary conditions of stable vortex formation in a frictionless medium.

According to the Standard Model, the electron is considered structureless. Yet, it exhibits angular momentum (spin), magnetic moment, and apparent internal oscillation—features typically associated with extended objects. Dirac’s equation in 1928 indicated the presence of internal motion at the speed of light, while experimental evidence suggested slight asymmetries in the electron’s charge distribution.

In the vortex model, the electron is a frictionless, irrotational vortex formed from vacuum condensation. The vortex has a circular core where the speed of rotation at every point is the speed of light, and the circulation ${\Gamma }_e=2\pi r_{e}c$ is constant. The momentum associated with this circulation yields:

${\Gamma }_e=2\pi r_{e}c,h={\Gamma }_e{m}_e,r_e=h/2\pi m_{e}c$ (5.4.1)

This provides a physical derivation for the Compton wavelength λ_C = 2πr = h/mc. The wave-like properties of the electron are tied to this structure:

The rotational frequency f = c/2πr matches the frequency derived from Planck’s relation E = hf, confirming the equivalence.

This model bridges the gap between classical mechanics and quantum theory, providing real geometry behind wave-particle duality. The de Broglie wavelength emerges from the translational motion, while the Compton wavelength arises from the internal rotation.

Building on Sbitnev’s quantum hydrodynamic framework, we incorporate a time-dependent oscillation of the vortex core radius to model fluctuations arising from vacuum elasticity and intrinsic quantum spinor dynamics. This dynamic trembling of the vortex core—interpreted as a localized manifestation of Zitterbewegung—can be described by:

$r_{core}\left(t\right)\approx 2\sqrt{\frac{a_0\nu }{\Omega }\cdot \frac{\sin \left(\Omega t\right)+n}{\Omega }}$ (6.1.1)

where:

This time-dependent radial modulation is consistent with Sbitnev’s generalized vorticity model, where quantum pressure and vacuum tension drive nonlocal dynamics. In our vortex framework, it reinforces the interpretation of the electron as a coherent structure in a superfluid-like vacuum background, with internal rotational structure superimposed on a de Broglie-guided translational path.

To incorporate these dynamics into a wave equation, we apply the appropriate energy operators to the structured wavefunction $\psi \left(r,\theta ,t\right)$:

where I = mr² is the moment of inertia of the vortex core.

Thus, the full vortex evolution equation becomes:

$i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2{\partial }^2\psi }{2m\partial z^2}-\frac{{\hslash }^2{\partial }^2\psi }{2I\partial {\theta }^2}+V\left(r,t\right)\psi$, (6.1.4)

where $V\left(r,t\right)$ may represent a potential term (external or self-consistent) depending on the vortex interaction with surrounding fields.

This formulation captures:

It bridges quantum field principles with hydrodynamic realism, opening a path for a physically grounded replacement of the Schrödinger equation rooted in vortex dynamics and superfluid vacuum theory.

The geometric structure inherent in the vortex model allows for the derivation of several critical physical quantities from classical and relativistic principles such as Lorentz factor, Bohr orbit circumference and fine-structure constant.

If the internal rotational speed perpendicular to the translation axis is v_⊥, and the total speed of the constituent elements of the vortex is c (the speed of light), the translational speed v can be related to v_⊥ and c.

The geometric structure of this model allows for the derivation of critical quantities. The rotational speed perpendicular to the translation axis is given by:

$v_{\perp }=\sqrt{c^2-v^2}$ (7.1.1.1)

This leads to a relativistic Lorentz factor:

$\gamma =\frac{c}{v_{\perp }}=\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{\frac{\sqrt{c^2-v^2}}{c}}=\frac{1}{\sqrt{\frac{c^2-v^2}{c^2}}}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ (7.1.1.2)

This geometric interpretation aligns with special relativity, showing how time dilation and internal structure are interconnected in the vortex framework.

Using vacuum permittivity ε₀ and known constants:

$v_0=\frac{e^2}{h{\epsilon }_0}\approx 2.1877\times {10}^6\text{m}/\text{s}$

$\lambda =\frac{h}{m{v}_0}\approx 3.32\times {10}^{-10}\text{m}$.

This matches the Bohr orbit circumference, corresponding to the ground state de Broglie wavelength in hydrogen.

For the central electron vortex:

$r=3.86\times {10}^{-13}\text{m}$

Circumference:

$C=2\pi r\approx 2.43\times {10}^{-12}\text{m}$

This aligns precisely with the Compton wavelength:

${\lambda }_C=h/mc\approx 2.426\times {10}^{-12}\text{m}$

The fine-structure constant α has long been regarded as one of the most mysterious dimensionless constants in physics. In the author’s previous work, “A New Theory on the Origin and Nature of the Fine Structure Constant” [25] , a novel explanation is proposed based on the vortex model. There, α is interpreted not as a fundamental constant arising from quantum electrodynamics, but as a geometric ratio intrinsic to vortex structures: specifically, the ratio between the rotational velocity at the boundary of the vortex and the speed of light at its center. This model reveals that the constancy of α emerges from the conserved circulation of an irrotational vortex in a superfluid vacuum, providing a natural and derivable origin for its numerical value.

Within this framework, the fine-structure constant appears as the ratio of Compton to de Broglie frequencies:

$\frac{f_c}{f_{dB}}\approx {\alpha }^{-1}=137$

This indicates that one de Broglie orbital cycle encompasses approximately 137 Compton-scale rotations—directly linking the structure of electron motion in the vortex model to the observed value of α, and grounding its constancy in vortex geometry (see Figure 1 ).

The derivation proceeds by considering the partial derivatives of the vortex wave function with respect to time and the spatial/angular coordinates.

Applying the energy operator, $i\hslash \partial /\partial t$, to the vortex wave function ψ yields:

$i\hslash \partial \psi /\partial t=i\hslash \left(-i\omega \right)\psi =\hslash \omega \psi$ (8.1.1.1)

this directly leads to:

$i\hslash \frac{\partial \psi }{\partial t}=E\psi$ (8.1.1.2)

the time-dependent Schrödinger equation where E represents the total energy of the system.

Applying the momentum operator along the z-direction, we analyze the translational component of the phase term in the wavefunction:

$\psi \left(r,t\right)=A\cdot \exp \left[i\left(\frac{2\pi z}{{\lambda }_{dB}}-\omega t+\frac{2\pi \theta }{{\lambda }_C}\right)\right]$ (8.1.2.1)

Let the translational phase term be expressed using de Broglie momentum:

$k_{dB}z=\frac{2\pi z}{{\lambda }_{dB}}=\frac{p_{z}z}{\hslash }$ (8.1.2.2)

Applying the operator $-i\hslash \frac{\partial }{\partial z}$:

$-i\hslash \frac{\partial \psi }{\partial z}=p_z\psi$ (8.1.2.3)

Squaring the operator for kinetic energy, we obtain:

$-\frac{{\hslash }^2}{\left(2m\right)\frac{{\partial }^2\psi }{\partial z^2}}=\frac{p_z^2}{2m}\psi =T_{trans}\psi$ (8.1.2.4)

Thus, the spatial part corresponds to standard kinetic energy in the z-direction.

The rotational phase term is given by:

$k_C\theta =2\pi \theta /{\lambda }_C=L_z\theta /\hslash$ (8.1.3.1)

Applying the angular momentum operator $-i\hslash \partial /\partial \theta$:

$-i\hslash \partial \psi /\partial \theta =L_{int}\psi$ (8.1.3.2)

where $L_{int}=mc{\lambda }_C=\hslash$.

The second derivative corresponds to the rotational kinetic energy operator:

$-{\hslash }^2/\left(2I\right){\partial }^2\psi /\partial {\theta }^2=\left(L_{int}^2/2I\right)\psi =T_{rot}\psi$ (8.1.3.3)

where I = mr² is the moment of inertia with r = λ_C.

Consider an ensemble of electrons, each described by a helical vortex wave function of the form:

${\psi }_j\left(r,\theta ,t\right)=A\cdot \exp \left[i\left(kz-\omega t+{\phi }_j\right)\right]$ (15.1.1)

where φ_j is the internal rotational phase associated with the vortex structure of the j-th particle. The total observed amplitude is the coherent sum of individual vortex contributions:

${\Psi }_{total}\left(r,t\right)={\displaystyle {\sum }_{j=1}^N\left(r,t\right)}=A\cdot {\displaystyle {\sum }_j\exp \left[i\left(kz-\omega t+{\phi }_j\right)\right]}$. (15.1.2)

The observed intensity, interpreted as probability density, is:

$P\left(r,t\right)={\left|{\Psi }_{total}\left(r,t\right)\right|}^2={\left|{\displaystyle {\sum }_{j=1}^N{\text{e}}^{i{\phi }_j}}\right|}^2$. (15.1.3)

If the internal phases φ_j are uniformly distributed (i.e., incoherent), the interference terms average out, and the result simplifies to:

$P\left(r,t\right)\propto N\cdot {\left|\psi \left(r,t\right)\right|}^2$ (15.1.4)

This shows that even though each particle follows a deterministic vortex trajectory, the probabilistic distribution observed across an ensemble can emerge from randomization of initial internal phases.

The vortex core may undergo rapid internal oscillations, similar to the Zitterbewegung effect. These radial oscillations can be modeled as:

$r\left(t\right)=r_0+\epsilon \cdot \sin \left(\Omega t+{\theta }_0\right)$ (15.2.1)

where r₀ is the average vortex radius, ε is the oscillation amplitude, Ω is the angular frequency, and θ₀ is the initial phase. Over time, this modulation causes the interaction point between the electron and measurement device to vary.

The long-term average detection probability becomes:

$P\left(r\right)\propto \underset{T\to \infty }{\lim }\frac{1}{T}\cdot {\displaystyle {\int }_0^T{\left|\psi \left(r\left(t\right)\right)\right|}^2\text{d}t}$ (15.2.2)

This result shows that even a single vortex, when sampled over time due to its internal motion, can exhibit a statistical behavior equivalent to the Born rule.

Assume that the vortex core explores a bounded region R in configuration space through its internal dynamics. If this motion is ergodic (i.e., it eventually visits all accessible states), then the fraction of time the vortex spends in a small volume element dV is given by:

$\frac{\tau \left(\text{d}V\right)}{T}\to {\displaystyle {\int }_{\text{d}V}{\left|\psi \left(r,t\right)\right|}^2{\text{d}}^{3}r}$ (15.3.1)

Thus, the probability of detecting the electron in region dV is proportional to the integral of the squared modulus of the wave function over that region—precisely the Born rule.

These three mechanisms—1) ensemble averaging over internal phases, 2) time-averaging over internal oscillations, and 3) ergodic exploration of space—suggest that the Born rule is not a fundamental randomness, but rather a statistical result of deterministic internal structure.

Future work should focus on:

Ultimately, this perspective aligns with the core philosophy of the vortex model: that quantum behavior arises from deep geometric and fluidic structures in the vacuum, rather than abstract axioms. The Born rule becomes not a mystery, but a mathematically emergent property of vortex-based reality.

Quantum entanglement, famously highlighted by the Einstein-Podolsky-Rosen (EPR) paradox, has posed profound philosophical challenges to deterministic interpretations of quantum mechanics. In standard quantum theory, entanglement describes a condition where the properties of two particles become interlinked, resulting in instantaneous correlations regardless of spatial separation. This phenomenon appears non-local and inherently probabilistic in the conventional interpretation.

In the vortex-based deterministic model presented here, entanglement is reinterpreted as a synchronization of internal vortex phases established during the initial interaction between particles. The internal structure and rotational dynamics of vortex entities can become phase-locked, analogous to classical coupled oscillators. Upon separation, these vortices remain correlated due to the conservation of internal angular momentum and phase coherence. Thus, what quantum mechanics describes as “non-locality” is, within this deterministic vortex framework, a preserved synchronization of phases embedded in the geometry of the vacuum medium.

Bell’s inequality represents a quantitative test distinguishing classical local realism from quantum mechanics, with experimental evidence favoring quantum predictions and thus suggesting non-local behavior. The vortex framework accommodates Bell-type correlations naturally through the inherent non-local characteristics of vortex fluid dynamics. Specifically, the quantum vacuum is treated as a superfluid medium permitting instantaneous phase adjustments over spatial distances, consistent with vortex hydrodynamics.

In this context, violation of Bell’s inequality does not imply faster-than-light signaling or true non-local interactions in the classical sense; instead, it reflects instantaneous coherence adjustments within a single, unified vortex-medium system. Hence, the vortex interpretation resolves Bell’s paradox by clarifying non-local correlations as natural outcomes of coherent, structured motion rather than probabilistic quantum leaps or mystical influences.

Quantum tunneling, another quintessential quantum paradox, is traditionally explained as a particle probabilistically overcoming a potential barrier through wavefunction penetration. Within the deterministic vortex model, tunneling emerges as a geometrical and hydrodynamical phenomenon. The vortex structure, characterized by internal angular momentum and coherent vacuum flow, interacts with potential barriers through structured resonance conditions. Instead of tunneling probabilistically, the vortex dynamically reconfigures its internal parameters (e.g., radius and angular velocity), enabling passage through or over barriers in a deterministic, coherent fashion.

This interpretation transforms tunneling from a probabilistic anomaly to a predictable resonance phenomenon dependent on internal vortex geometry and vacuum medium interactions.

In standard quantum mechanics, measurement-induced wavefunction collapse poses an interpretational challenge, attributing special status to the observer. The vortex-based model eliminates the necessity for collapse altogether. Measurement outcomes emerge deterministically from interactions between vortices and measurement apparatuses, determined by initial conditions and internal vortex configurations.

Thus, the so-called collapse is merely a selection or resonance with specific vortex phases. Probabilistic outcomes represent averages over many deterministic vortex interactions, effectively demystifying the measurement problem by reestablishing physical continuity and causality.

The uncertainty principle sets fundamental limits on simultaneous precision of complementary variables (position and momentum). Within the deterministic vortex framework, uncertainty reflects practical measurement limitations rather than intrinsic indeterminism. The complex internal dynamics and finite scale of vortex structures inherently restrict simultaneous precision, reflecting classical limitations of measuring extended, rotating bodies rather than fundamental randomness.

In conclusion, classical quantum paradoxes—including entanglement, Bell’s inequality, tunneling, wavefunction collapse, and uncertainty—are explicitly and coherently resolved in this deterministic vortex model. Each paradox transforms from a fundamental mystery into a comprehensible, causal, and geometrically grounded feature of vacuum vortex dynamics, fully aligning with empirical quantum outcomes.