Math Summary: Gravity Interaction Model

BeeTheory Math Summary: Gravity Interaction Model

We consider two elementary particles $ A_0 $ and $ B_0 $ modeled by wave functions that we sum:

\[ \Psi(x, y, z, t) = \Psi(A, t) + \Psi(B, t) \]

\[ \Psi(x, y, z, t) = A \cdot e^{-\alpha(\{x, y, z\} – A_0)} \cdot e^{i\omega_1 t} + B \cdot e^{-\beta(\{x, y, z\} – B_0)} \cdot e^{i\omega_2 t} \]

We change the frame of reference to spherical coordinates:

\[ \Psi(R, t) = A \cdot e^{-\alpha(R_A-A_0)} \cdot e^{i\omega_1 t} + B \cdot e^{-\beta(R_B-B_0)} \cdot e^{i\omega_2 t} \]

The positions of particles $ A_0 $ and $ B_0 $ are considered fixed at the considered time scale. We focus around the second particle $ B_0 $:

\[ \Psi(R, t) = \Psi(R_B + r, t) \]

\[ R_A = R_{A0B0} + r, \quad R_B = r, \quad r \text{ is small}. \]

\[ \Psi(R, t) = A \cdot e^{-\alpha(R_{A0B0} + r)} \cdot e^{i\omega_1(t+d_1)} + B \cdot e^{-\beta r} \cdot e^{i\omega_2(t+d_2)} \]

We apply the Schrödinger equation, considering that there is only kinetic energy and no potential energy. $ V $ is null everywhere.

\[ i\hbar \frac{\partial}{\partial t} \Psi(R,t) = T + V = T \]

\[ i\hbar \frac{\partial}{\partial t} \Psi(R,t) = -2m\hbar^2 \nabla^2 \Psi(R, t) \]

Positioning ourselves at $ B_0 $, we simplify by calculating only the first term related to $ A $, the term related to $ B $ is null at $ B_0 $; we extract the term in $ R_{A0B0} $ which is a constant:

\[ i\hbar \frac{\partial}{\partial t} \Psi(R,t) = -2m\hbar^2 \nabla^2(A e^{-\alpha R_{A0B0}} \cdot e^{-\alpha \cdot r/R_{A0B0}}) \]

Using the Laplacian in spherical coordinates for a function that depends only on $ r $:

\[ \nabla^2 f(r) = \frac{1}{r^2} \frac{d}{dr} (r^2 \frac{df}{dr}) \]

\[ \nabla^2 f(r) = \frac{1}{r^2} \frac{d}{dr} (r^2 \cdot \frac{d}{dr} e^{-\alpha \cdot r/R_{A0B0}}) \]

\[ r^2 \cdot \frac{d}{dr} \psi(r) = r^2 \cdot \frac{d}{dr} (e^{-\alpha r/R_{A0B0}}) = r^2 \cdot (-\alpha r/R_{A0B0}) \cdot e^{-\alpha r/R_{A0B0}} \]

\[ \nabla^2 f(r) = \frac{1}{r^2} \frac{d}{dr}(r^2 \cdot -\alpha r/R_{A0B0} \cdot e^{-\alpha r/R_{A0B0}}) \]

\[ \nabla^2 f(r) = \frac{1}{r^2} \cdot -\alpha/R_{A0B0} \cdot \frac{d}{dr}(r^3 \cdot e^{-\alpha r/R_{A0B0}}) \]

Recalling that $ R_{A0B0} $ is large and $ r $ is very small:

\[ \nabla^2 f(r) \approx -3\alpha/R_{A0B0} \]

Therefore, we obtain a potential proportional to the inverse of the distance between the particles.

In the realm of quantum mechanics, the description of particles as wave functions represents a fundamental shift from classical physics, which typically treats particles as discrete entities with definite positions and velocities. This conceptual transition to wave-particle duality allows for a more comprehensive understanding of the behavior of subatomic particles, such as electrons and photons, particularly in terms of their interactions, propagation, and the effects of confinement on their quantum states.

Quantum mechanics posits that every particle is associated with a wave function, which provides a probabilistic description of its quantum state as a function of position and time. The wave function, often denoted as Ψ (Psi), encapsulates all the information about a particle’s quantum state and is fundamental to predicting how that state evolves over time according to the Schrödinger equation.

This introduction delves into the mathematical modeling of wave functions for two elementary particles, exploring their sum and interactions through a comprehensive mathematical framework. These particles are modeled in a way that allows us to examine their dynamics under various transformations, such as coordinate system changes, and interactions within the framework of non-relativistic quantum mechanics.

Mathematical Representation of Wave Functions

The standard form of a wave function for a particle in quantum mechanics is complex-valued, incorporating both an amplitude and a phase. This function is a solution to the Schrödinger equation, which describes how the wave function evolves in space and time. The equation is linear, allowing for the superposition of solutions, which means that if two wave functions are solutions, their sum is also a solution. This principle underlies our approach to modeling interactions between particles using their respective wave functions.

Modeling Particle Interactions

For our model, we consider two particles, designated as $𝐴_{0}$ and $𝐵_{0}$ , each described by its wave function. The overall system is then described by the superposition of these wave functions, leading to a combined wave function that provides a field of probability amplitudes. Analyzing these superpositions helps us understand how particles influence each other’s quantum states through phenomena such as interference and entanglement.

Transition to Spherical Coordinates

In the analysis of quantum systems, choosing an appropriate coordinate system can significantly simplify the mathematical treatment, especially when dealing with spherically symmetric systems such as atoms or spherical potential wells. By transitioning to spherical coordinates, we can more effectively describe the radial dependencies and angular momentum properties of the system. This coordinate transformation is crucial when the natural symmetry of the physical system aligns with spherical coordinates, which is often the case in atomic and molecular systems.

Focus on Kinetic Energy

In our model, we assume that the potential energy $𝑉$ is null, which implies that we are focusing solely on the kinetic energy component of the quantum system. This simplification is common in theoretical treatments of free particles or for illustrating fundamental quantum mechanics concepts without the complicating factors of potential energies. The kinetic energy operator, denoted as $𝑇$ , then becomes the primary driver of the dynamics described by the wave function.

Advanced Mathematical Techniques

The use of advanced mathematical techniques such as the Laplacian in spherical coordinates becomes indispensable in our analysis. These techniques allow us to delve into the differential aspects of the wave function, providing insights into how changes in the spatial configuration of the system influence the behavior of the particles. The Laplacian operator, in particular, plays a key role in determining how the wave function’s amplitude and phase evolve in space, which is directly related to the observable properties of the system such as the distribution of positions and momenta.

In conclusion, this introduction sets the stage for a detailed exploration of the quantum mechanical modeling of particle interactions. By examining the superposition of wave functions and the application of the Schrödinger equation in a context devoid of potential energy, we aim to uncover the nuanced dynamics of elementary particles in a purely kinetic framework, thus enriching our understanding of quantum mechanics and its foundational principles.

Let’s break down the key components and summarize the mathematical progression:

1. Wave Function Representation

Two particles, $A_0$ and $B_0$ , are modeled by their wave functions:

$\Psi(x, y, z, t) = A e^{-\alpha(\{x, y, z\} – A_0)} e^{i\omega_1 t} + B e^{-\beta(\{x, y, z\} – B_0)} e^{i\omega_2 t}.$

This representation assumes:

Amplitude Terms ( $A, B$ ) and spatial decay ( $e^{-\alpha r}, e^{-\beta r}$ ).
Oscillatory Time Dependence ( $e^{i\omega t}$ ) characteristic of quantum states.

2. Change to Spherical Coordinates

Switching to spherical coordinates simplifies the analysis of radial dependencies, particularly when studying localized interactions around one particle (e.g., $B_0$ ):

$\Psi(R, t) = A e^{-\alpha(R_{A_0B_0} + r)} e^{i\omega_1(t+d_1)} + B e^{-\beta r} e^{i\omega_2(t+d_2)}.$

Here:

$R_{A_0B_0}$ : The fixed distance between particles $A_0$ and $B_0$ .
$r$ : The small deviation from $B_0$ .

3. Schrödinger Equation Application

Assuming no potential energy ( $V = 0$ ), the kinetic energy operator ( $T$ ) governs the evolution of the wave function:

$i\hbar \frac{\partial}{\partial t} \Psi(R, t) = -\frac{\hbar^2}{2m} \nabla^2 \Psi(R, t).$

Focusing on the contribution from $A$ , the spatial term simplifies to:

$\Psi(R, t) \sim A e^{-\alpha R_{A_0B_0}} e^{-\alpha \frac{r}{R_{A_0B_0}}}.$

4. Laplacian in Spherical Coordinates

Using the Laplacian operator for radially dependent functions:

$\nabla^2 f(r) = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} f(r) \right),$

we calculate:

$f(r) = e^{-\alpha \frac{r}{R_{A_0B_0}}}.$

Steps:

Compute $r^2 \frac{\partial}{\partial r}$ : $r^2 \frac{\partial}{\partial r} \left( e^{-\alpha \frac{r}{R_{A_0B_0}}} \right) = r^2 \left( -\frac{\alpha}{R_{A_0B_0}} e^{-\alpha \frac{r}{R_{A_0B_0}}} \right).$
Differentiate again: $\nabla^2 f(r) \approx -\frac{3\alpha}{R_{A_0B_0}}.$

5. Emergent Inverse Distance Potential

The Laplacian reveals that the wave function generates a term proportional to $\frac{-1}{R_{A_0B_0}}$ , implying an effective potential inversely proportional to the distance between particles. This suggests that gravitational or interaction-like effects naturally emerge from the quantum wave function formalism.

Key Physical Insights

Wave Function Interactions: The superposition principle allows modeling particle interactions, where interference patterns encode information about their relative positions and dynamics.
Kinetic Energy Dominance: Assuming no potential energy focuses the analysis purely on the spatial and temporal evolution driven by kinetic terms.
Gravitational Analogy: The appearance of an inverse-distance term in the wave function behavior hints at a quantum foundation for gravitational-like interactions, where wave properties govern long-range effects.

Future Directions

Incorporating Potential Energy: Adding a potential $V(r)$ could refine the model, capturing external forces or fields acting on the particles.
Relativistic Corrections: For a complete quantum-gravitational framework, extending to relativistic wave equations (e.g., Klein-Gordon or Dirac equations) may be necessary.
Entanglement and Non-Locality: Examining how wave functions influence each other could explore entanglement or non-local interaction mechanisms in gravity.

This mathematical framework provides a stepping stone for understanding quantum interactions with a gravitational interpretation, potentially bridging quantum mechanics and classical gravity.