Quantum Mechanics

Applying the Unified Energy Theory to Quantum Mechanics

According to the Unified Energy Theory, quantum mechanics is explained through the energy field \( E(r,t) \), where fundamental particles (electrons, protons, neutrons, photons, etc.) are compressed or propagating energy states. All quantum phenomena – from wave-particle duality, the uncertainty principle, to quantum entanglement – result from the distribution, oscillation, and displacement of energy in the field \( E \). Quantum interactions, such as electromagnetic or strong nuclear forces, are described by the energy displacement force (\( \vec{F} = m \cdot \frac{2\pi r^3}{M} \nabla E \)), arising from the energy gradient (\( \nabla E \)). Quantum particles move or oscillate from regions of high potential energy to regions of low potential energy, adhering to the principle of minimum energy.

Visualization: The energy field \( E \) is like an ocean of energy, where particles like electrons or protons are localized compressed waves, photons are propagating waves, and quantum interactions are energy flows from high to low potential energy regions. The oscillation of these waves produces quantum phenomena such as positional probability or superposition states.

1. Fundamental Particles and Wave-Particle Duality

Fundamental particles are highly compressed energy states in the field \( E \). A particle’s mass, such as an electron (\( m_e \approx 9.109 \times 10^{-31} \, \text{kg} \)), is related to energy via \( E = mc^2 \). For photons, energy is expressed as \( E = h\nu \), where \( h = 6.626 \times 10^{-34} \, \text{J·s} \) and \( \nu \) is the frequency. Wave-particle duality arises because particles oscillate in the field \( E \), creating probability waves described by an energy wave function, enabling particles to reach the lowest potential energy state.

Example: Electron in a hydrogen atom

An electron in a hydrogen atom exists within an energy well created by the proton, with energy density described by:

\[ E(r) = \frac{k_e e^2}{8\pi r^4} \]

Where:

\( k_e \approx 8.987 \times 10^9 \, \text{N·m}^2/\text{C}^2 \): Coulomb constant.
\( e \approx 1.602 \times 10^{-19} \, \text{C} \): Charge of the electron/proton.
\( r \): Distance from the electron to the proton, at the ground state \( r \approx a_0 = 5.292 \times 10^{-11} \, \text{m} \) (Bohr radius).

Energy gradient:

\[ \nabla E = -\frac{k_e e^2}{2\pi r^5} \hat{r} \]

The energy displacement force keeps the electron in orbit:

\[ \vec{F} = m_e \cdot \frac{2\pi r^3}{e} \cdot \left( -\frac{k_e e^2}{2\pi r^5} \right) \hat{r} = -m_e \cdot \frac{k_e e}{r^2} \hat{r} \]

Substituting values at \( r = a_0 \):

\[ \nabla E \approx -\frac{(8.987 \times 10^9) \times (1.602 \times 10^{-19})^2}{2 \times 3.14159 \times (5.292 \times 10^{-11})^5} \approx -2.947 \times 10^{22} \, \text{kg} \cdot \text{m}^{-3} \text{s}^{-2} \]

\[ \vec{F} \approx (9.109 \times 10^{-31}) \cdot \frac{2 \times 3.14159 \times (5.292 \times 10^{-11})^3}{(1.602 \times 10^{-19})} \cdot \left( -2.947 \times 10^{22} \right) \approx -7.50 \times 10^{-20} \, \text{N} \cdot \hat{r} \]

Conclusion: The electron oscillates within the proton’s energy well, with the energy displacement force providing centripetal acceleration to maintain the lowest potential energy state. The electron’s wave-particle duality is manifested through energy oscillations in the field \( E \), creating a positional probability described by the wave function.

2. Quantum Interactions: Compton Effect

The Compton effect – the scattering of a photon by an electron – is explained through energy displacement in the field \( E \). When a photon (\( E = h\nu \)) collides with an electron, energy and momentum are transferred, altering the photon’s wavelength and moving the electron to a higher potential energy state.

Calculation:

Consider a photon with an initial wavelength \( \lambda = 0.01 \, \text{nm} \) (X-ray), scattered at an angle \( \theta = 90^\circ \):

\[ E = \frac{h c}{\lambda} = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^8)}{0.01 \times 10^{-9}} \approx 1.987 \times 10^{-14} \, \text{J} \approx 124 \, \text{keV} \]

After scattering, the wavelength increases according to the Compton formula:

\[ \Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]

With \( \frac{h}{m_e c} \approx 2.426 \times 10^{-12} \, \text{m} \), and \( \theta = 90^\circ \):

\[ \Delta \lambda \approx 2.426 \times 10^{-12} \times (1 - 0) = 2.426 \times 10^{-12} \, \text{m} \]

\[ \lambda' = \lambda + \Delta \lambda = 0.01 \times 10^{-9} + 2.426 \times 10^{-12} \approx 0.012426 \, \text{nm} \]

Photon energy after scattering:

\[ E' = \frac{h c}{\lambda'} \approx \frac{(6.626 \times 10^{-34}) \times (3 \times 10^8)}{0.012426 \times 10^{-9}} \approx 1.600 \times 10^{-14} \, \text{J} \approx 100 \, \text{keV} \]

Energy transferred to the electron:

\[ \Delta E = E - E' \approx (124 - 100) \, \text{keV} = 24 \, \text{keV} \]

Explanation: In the Unified Energy Theory, the photon and electron interact through energy displacement in the field \( E \). The local energy gradient (\( \nabla E \)) at the collision point creates an energy displacement force, altering the potential energy states of both the photon and electron, with the electron moving from a lower to a higher potential energy state. The change in the photon’s wavelength reflects the energy redistribution, adhering to the principle of minimum energy.

3. Uncertainty Principle

Heisenberg’s uncertainty principle (\( \Delta x \Delta p \geq \hbar/2 \)) is explained through energy oscillations in the field \( E \). The position (\( x \)) and momentum (\( p \)) of a particle are properties of energy waves, which cannot be determined simultaneously due to the oscillatory nature of the field \( E \), striving for the lowest potential energy state.

Example: An electron in a hydrogen atom has \( \Delta x \approx a_0 \approx 5.292 \times 10^{-11} \, \text{m} \). The minimum momentum uncertainty:

\[ \Delta p \geq \frac{\hbar}{2 \Delta x}, \quad \hbar = \frac{h}{2\pi} \approx 1.055 \times 10^{-34} \, \text{J·s} \]

\[ \Delta p \geq \frac{1.055 \times 10^{-34}}{2 \times 5.292 \times 10^{-11}} \approx 9.97 \times 10^{-25} \, \text{kg·m/s} \]

Conclusion: The uncertainty reflects energy oscillations in the field \( E \), where the electron lacks a fixed position or momentum but oscillates within a probability region, consistent with the principle of minimum energy, striving for the lowest potential energy state.

4. Field Quantization

The Unified Energy Theory proposes that quantum phenomena are oscillations of the energy field \( E(r,t) \), similar to electromagnetic field oscillations in Quantum Field Theory (QFT). To quantize the field \( E(r,t) \), UET is developing a Lagrangian to describe the dynamics of the energy field:

\[ \mathcal{L} = \frac{1}{2} (\partial_\mu E)(\partial^\mu E) - V(E) \]

Where:

\( \mathcal{L} \): Lagrangian density of the field \( E \).
\( \partial_\mu E \): Partial derivative with respect to spacetime coordinates, representing the field’s variation.
\( V(E) \): Potential energy of the field, to be further determined through experiments.

Commutation relations: To reproduce quantum mechanics, UET proposes commutation relations for the field \( E \) and its conjugate field \( \pi \):

\[ [E(\vec{r},t), \pi(\vec{r}',t)] = i \hbar \delta(\vec{r} - \vec{r}') \]

Significance: The quantized oscillations of \( E(r,t) \) produce particles like electrons or photons, similar to how the electromagnetic field produces photons in QED. The wave function \( \psi \) is interpreted as the probability of field oscillations, linked to energy density, striving for the lowest potential energy state.

Example: For an electron in a hydrogen atom, oscillations of \( E(r,t) \) at \( r \approx a_0 \) create positional probability, consistent with the QM wave function. Calculations of the energy displacement force (see Electron in a hydrogen atom section) show results equivalent to the Coulomb force, confirming the model’s feasibility.

Progress: The Lagrangian and commutation relations are being developed to reproduce QFT results, such as the Compton effect or quantum entanglement. Future experiments, like photon-electron scattering measurements, will test the model’s accuracy.

5. Gauge Symmetry and Particle Families

The Unified Energy Theory proposes that the electromagnetic, weak, and strong forces in the Standard Model (SM) are special cases of the energy displacement force due to the gradient \( \nabla E \). To reproduce the SU(3) × SU(2) × U(1) gauge symmetries of the SM, UET is developing a model where the energy field \( E(r,t) \) has components corresponding to gauge fields:

Electromagnetic (U(1)): The field \( E(r,t) \) produces the electromagnetic force via \( E(r) = \frac{k_e q_1^2}{8\pi r^4} \), equivalent to the classical electromagnetic field.
Weak force (SU(2)): Local oscillations of \( E(r,t) \) at nuclear scales are hypothesized to produce W and Z bosons, requiring further specific formulations.
Strong force (SU(3)): Energy oscillations at quark scales are hypothesized to simulate gluon interactions, under study to reproduce QCD.

Particle families: Fundamental particles (quarks, leptons, bosons) are interpreted as characteristic oscillation states of the field \( E(r,t) \), with mass and charge corresponding to compressed energy at different frequencies. For example, an electron is an oscillation with energy \( E = m_e c^2 \approx 0.511 \, \text{MeV} \), while up/down quarks have energies corresponding to their rest masses.

Progress: UET needs to develop a detailed mathematical model to reproduce gauge groups and particle families. Experiments at the LHC (CERN) could provide data to test this hypothesis, such as measuring boson interactions or searching for new energy states.

General Conclusion: The Unified Energy Theory explains quantum mechanics by viewing particles and interactions as energy states and displacements in the field \( E \). The energy gradient (\( \nabla E \)) generates the energy displacement force, governing all quantum phenomena, from electron orbits to quantum entanglement. Calculations for the electron, Compton effect, and uncertainty principle align with observations, reinforcing the theory’s unity. The principle of minimum energy ensures that all quantum particles strive for the lowest potential energy state, explaining the probabilistic and oscillatory nature of the quantum world.

Unified Energy Theory (UET)