An introduction to Spiral Wave Theory

Exploring the Spiral Wave Theory: A New Perspective on Quantum Mechanics

When we observe the movement of celestial bodies from an astronomical perspective, we often describe the orbits of planets as elliptical paths around stars. However, considering that our universe is constantly expanding, these elliptical paths are actually part of a larger spiral trajectory through spacetime. This spiral motion can be seen not only in the macroscopic scales of planets and stars but also in the microscopic behavior of particles and waves. By re-examining the laser slit experiments and diffraction patterns through the lens of spiral wave theory, we may find a simpler explanation for the observed interference patterns—without the need to invoke a quantum universe where particles exist in multiple places simultaneously.

1. Introduction to Spiral Wave Theory

The traditional view in quantum mechanics is that particles follow straight-line paths or probabilistic wave-like trajectories. The spiral wave theory challenges this notion by suggesting that particles follow spiral paths in both space and time. This concept can potentially offer new insights into quantum behaviors and the nature of forces.

2. Mathematical Development

To explore the spiral wave theory, we start by defining a spiral trajectory in cylindrical coordinates (r, θ, z), where the z-coordinate represents the spiral's axial progression:

r = r
θ = θ
z = bθ

Here, b is a constant defining the pitch of the spiral. We then modify the time-dependent Schrödinger equation in three dimensions to incorporate these spiral paths.

3. Modified Schrödinger Equation

The time-dependent Schrödinger equation is given by:

iħ ∂ψ(𝐫, t)/∂t = -ħ²/2m ∇² ψ(𝐫, t) + V(𝐫, t) ψ(𝐫, t)

In cylindrical coordinates, the Laplacian ∇² is:

∇² = 1/r ∂/∂r (r ∂/∂r) + 1/r² ∂²/∂θ² + ∂²/∂z²

Substituting z = bθ, we get:

∂/∂z = 1/b ∂/∂θ
∂²/∂z² = 1/b² ∂²/∂θ²

Thus, the Laplacian in spiral coordinates becomes:

∇² = 1/r ∂/∂r (r ∂/∂r) + (1 + 1/b²) 1/r² ∂²/∂θ²

We can then substitute this into the Schrödinger equation and separate variables to obtain differential equations for the radial and angular parts.

4. Solving the Equations

Radial Equation with Harmonic Oscillator Potential

We use the harmonic oscillator potential V(r) = ½ m ω² r² and solve the resulting radial equation numerically.

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Parameters for the harmonic oscillator potential
m = 1  # mass (arbitrary units)
omega = 1  # angular frequency (arbitrary units)
hbar = 1  # reduced Planck's constant (arbitrary units)
r_max = 10  # maximum value of r to solve for

# Differential equation for the radial part
def radial_eq(r, R):
    dR_dr = R[1]
    d2R_dr2 = (m² * omega² / hbar²) * r³ * R[0] - (2 * m * hbar * omega / hbar²) * r * R[0]
    return [dR_dr, d2R_dr2]

# Boundary conditions: R(0) should be finite, R(r_max) should be 0
R0 = [1, 0]  # initial guess for R(0) and R'(0)

# Solve the radial equation
r_span = (0, r_max)
r_eval = np.linspace(0, r_max, 1000)
solution = solve_ivp(radial_eq, r_span, R0, t_eval=r_eval)

# Extract the solution
r_values = solution.t
R_values = solution.y[0]

# Plot the radial solution
plt.figure(figsize=(10, 6))
plt.plot(r_values, R_values, label='Radial Solution $R(r)$')
plt.xlabel('$r$')
plt.ylabel('$R(r)$')
plt.title('Radial Solution for the Spiral Wave Model with Harmonic Oscillator Potential')
plt.legend()
plt.grid(True)
plt.show()

Angular Equation

For the angular part, we assume a Fourier series solution and solve:

Θ(θ) = ∑ cₙ e^{inθ}

5. Complete Wavefunction

By combining the radial and angular solutions, we obtain the complete wavefunction:

ψ(r, θ, t) = R(r) (A cos(√k θ) + B sin(√k θ)) e^{-iωt}

6. Visualization

Finally, we visualize the complete wavefunction in a 3D plot.

from mpl_toolkits.mplot3d import Axes3D

# Constants for the angular part
A = 1
B = 1
r = r_values  # Using the solved r values from the radial equation

# Calculate k based on the average r value
k = (m * omega² * np.mean(r_values)⁴ / 2 / (1 + 1/b²) - np.mean(r_values)² * hbar * omega / (1 + 1/b²))

# Generate theta values
theta_values = np.linspace(0, 4 * π, 1000)  # 2 full turns

# Calculate Theta(θ)
Theta_values = A * cos(√k * theta_values) + B * sin(√k * theta_values)

# Create 3D plot
fig = plt.figure(figsize=(10, 6))
ax = fig.add_subplot(111, projection='3d')

# Calculate z-values for the 3D plot
z_values = b * theta_values

# Plot the spiral wave
ax.plot(r_values * cos(theta_values), r_values * sin(theta_values), z_values, label='Spiral Path')
sc = ax.scatter(r_values * cos(theta_values), r_values * sin(theta_values), z_values, c=np.abs(Theta_values), cmap='viridis', label='Wave Intensity')
fig.colorbar(sc, label='Wave Amplitude')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.setTitle('3D Spiral Wave Model')
ax.legend()
plt.show()

Conclusion

Exploring the spiral wave theory offers a fascinating new perspective on quantum mechanics. While this theory still needs further work, I believe it is one that merits investigation. If I had the funding and time, I could focus on these efforts.