# Is it Really Possible to Overcome the Wave-Particle Duality?

Undoubtedly, at the dawn of __ Quantum Theory__ birth there is the comparison between the

__'s__

**David Bohm**__, with hidden variables, and the__

**Deterministic Interpretation**__and__

**SchrĂ¶dinger**__, which introduces the__

**Born Probabilistic Paradigm**__,__

**Wavefunction**__as a__

**đťś“****representation**of a

**physical system**'s

**state**(

__Figure 1__).

## Towards a Unified Quantum Mechanics Theory

In the last years, a **Unified Quantum Mechanics** framework has been proposed, consisting of a comprehensive analysis of the **Wave**â€“**Particle Dynamics** of **quantum particles**. The conventional wavefunction, [math]\Large{\psi}[/math] (__Figure 2__) is redefined in terms of **real scalar** [math]\Large{{R}_0}[/math] and **vector** [math]\Large{\vec{R}}[/math] functions, shedding light on the **intrinsic nature** of quantum particles. The resulting expression unveils a dual composition of massless and massive fields within the particle, with the scalar [math]\Large{{R}_0}[/math] and vector [math]\Large{\vec{R}}[/math] functions satisfying the **Quantum Telegraph Equation** (__Equation 1__) [__1__] and the **phase**, [math]\Large{S}[/math] representing a massless field.

The interaction between these three functions is investigated, revealing intriguing connections. Notably, the **product** [math]\Large{S \cdot {R_0}}[/math] satisfies the telegraph equation, representing the potential energy of the particle field, while [math]\Large{S \cdot {\vec{R}}}[/math] signifies its **momentum**. This article explores scenarios where the particle exhibits **fluid-like (wave) behavior**, characterized by a conserved **Energyâ€“Momentum Tensor**, [math]\Large{{\sigma_{ij}}^m}[/math] (__Equation 2__).

**Equation 1**. The mathematical general form of the Quantum Telegraph Equation

Furthermore, **Bohmian Mechanics** can be incorporated into the __ Dirac Equation__, resulting in a relativistic

__and a__

**Hamilton â€“ Jacobi Equation**__. The new formalism eliminates the conventional quantum potential energy, providing a fresh perspective on__

**Continuity Equation****Quantum Mechanics**. Application of the Bohmian Method to

__'s__

**Maxwell**__demonstrates the__

**Equations****Particleâ€“Wave Duality**[

__2__] of the electromagnetic field. It can also be established that when the electromagnetic field behaves as matter, it resembles a massless Dirac particle, supporting the

**de Broglie Hypothesis**[

__3__].

**Equation 2**. The Energy-Momentum Tensor mathematical expression

where [math]\vec{R}_i[/math] and [math]\vec{R}_j[/math] are the [math]R[/math] **vector components** along the [math]i[/math] and [math]j[/math] **arbitrary directions**, respectively; while [math]\delta_{ij}[/math] is the **Delta Function**, which modulates the
tensor magnitude.

## What about a New Potential Energy?

In the so-called **SchrĂ¶dinger-Bohm Mechanics** [__4__], the motion of a non-relativistic quantum particle in a **Potential Energy**, [math]\Large{V}[/math] is described using the __ SchrĂ¶dinger Equation__. The wavefunction is expressed as:

**Equation 3**. One of the Electron Wavefunction scalar forms

where [math]\Large{R}[/math] and [math]\Large{S}[/math] are real functions. The resulting equations are analyzed, revealing the dual nature of the particle as both wave and matter. A **new potential energy dependent** on the **phase** [math]\Large{S}[/math] is introduced, termed the **Phase (Spin) Potential Energy**.

## A hybrid Theory of Vector and Scalar Wavefunctions

The **Vector Quantum Mechanics** represents a **free quantum particle** with mass, [math]\Large{m}[/math] using **scalar** and **vector wavefunctions** [math]\Large{\psi{_0}}[/math] and [math]\Large{\vec{\psi}}[/math], respectively. The quantum telegraph equation governs their evolution, illustrating the **fluid-like motion** [__5__] of the particle.

**Figure 1**. A representative 3-Dimensional Electron Wavefunction Energy Distribution graph

## From Bohmian Vector Quantum Mechanics to Relativistic Dirac Equation

The combination of Bohmian Vector Quantum Mechanics to vector quantum mechanics yields to a **relativistic Dirac Hamiltonian** for **massless** and **massive particles**. The Hamiltonian equality implies a dual description of the particle, with all information propagating at the **speed of light**. The mass of the particle is uniquely determined by its wavefunctions, showcasing the **interconnectedness of the fields**.

## Application to Maxwellâ€™s Equations

The Bohmian approach is further extended to Maxwell's equations, revealing the particleâ€“wave duality of the **Electromagnetic Field**. The electromagnetic field is expressed as a wave and particle, with the phase [math]\Large{S}[/math] governing the particle dynamics and [math]\Large{\vec{E}_0}[/math] and [math]\Large{\vec{B}_0}[/math] (**Electric Field** and **Magnetic Field** in **vacuum**, respectively, interacting with particle) representing wave aspects. The **Photon**, when behaving as matter, resembles a **massless Dirac particle**, supporting the de Broglie hypothesis.

**Figure 2**. A fountain plan of Electron Wavefunctions Interference

## A Delicious Chance to Connect the Matter and Wave features of Particles

The **Unified Quantum Mechanics** framework highlights the **inherent duality** and **interconnectedness** of massless and massive fields within quantum particles. Particularly, the **elimination** of the **quantum potential energy** in this **new formalism** opens avenues for further exploration in quantum theory.

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Journey into the Core Concepts"
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