Fractal wave functions aim to redefine the classical wave function concept of quantum mechanics with fractal geometry. In this approach, multiscale, self-similar structures stand out rather than the continuity of the wave function.

Basic Framework

• Fractal Hilbert Space: The expansion of the space where wave functions are defined into fractal dimensions.
• Multifractal Probability Density: The probability of finding the particle showing a scale-dependent distribution rather than a uniform one.
• Fractal Phase and Amplitude: Modeling the phase and amplitude components of the wave function with fractal dependencies.
• Fractal Schrödinger Equation: Rewriting the classical Schrödinger equation with fractal derivatives and fractional operators.

Application Areas

• Quantum Chaos: Explaining wave functions in chaotic systems with fractal structures.
• Nanostructures: Modeling the movement of electrons on fractal surfaces.
• Cosmology: Defining the wave function of the universe with fractal metrics.
• Models of Consciousness: Fractal explanation of mental processes through wave functions.

Example of Mathematical Formulation

The fractal wave function can be written as follows:

Ψ(𝑥, 𝑡) = 𝐴(𝑥, 𝑡) ⋅ 𝑒 ^{i⋅𝜙 (𝑥,𝑡)}

Here:

𝐴(𝑥, 𝑡) → multifractal amplitude distribution

𝜙(𝑥, 𝑡) → fractal phase function

Amplitude and phase are defined by the Hurst exponent and fractional derivatives instead of classical linear functions.

Conclusion

Fractal wave functions are a critical step for a unified model that combines quantum mechanics and fractal statistics.

Fractal Wave Functions – Mathematical Details

Fractal wave functions expand the linear structure of classical quantum mechanics with fractional derivatives and multifractal distributions. The goal here is to define both the amplitude and phase components of the wave function with fractal dependencies.

Basic Structure

Fractal Amplitude Function

𝐴(𝑥, 𝑡) ∼∣ 𝑥 ∣^𝐻 ⋅ 𝑓(𝑡)

Here 𝐻 → Hurst exponent, determines the scale dependence of the amplitude.

Fractal Phase Function

𝜙(𝑥, 𝑡) = 𝛼 ⋅ 𝑥^𝐷_f + 𝛽 ⋅ 𝑡^𝐷_f

The phase scales with the fractal dimension 𝐷_f.

Fractal Wave Function

Ψ(𝑥, 𝑡) = 𝐴(𝑥, 𝑡) ⋅ 𝑒 ^{i⋅𝜙 (𝑥,𝑡)}

Redefinition of the classical wave function with fractal amplitude and phase.

Fractal Schrödinger Equation

Classical Schrödinger equation:

𝑖ℏ (𝜕/𝜕𝑡) Ψ(𝑥, 𝑡) = − (ℏ²/2𝑚)∇² Ψ(𝑥, 𝑡) + 𝑉(𝑥)Ψ(𝑥, 𝑡)

In its fractal version, the derivatives are written as fractional derivatives:

𝑖ℏ (𝜕^𝛼/𝜕𝑡^𝛼) Ψ(𝑥, 𝑡) = − (ℏ²/2𝑚)∇^2𝐷_f Ψ(𝑥, 𝑡) + 𝑉(𝑥)Ψ(𝑥, 𝑡)

𝛼 → temporal fractal order

𝐷_f  → spatial fractal dimension

Statistical Properties

Multifractal Probability Density

𝑃(𝑥) ∼∣ 𝑥 ∣^-𝜇

Here 𝜇 determines the fractal distribution of the particle’s probability of being found.

Fractal Entropy

𝑆_𝑞 = ( 1 − ∑_i 𝑝_i ^𝑞 ) / (𝑞 − 1)

The adapted form of Tsallis entropy with fractal amplitudes.

Application Perspective

• Quantum Chaos: Wave functions exhibiting chaotic behavior with fractal dimensions.
• Nanomaterials: Movement of electrons on fractal surfaces.
• Cosmology: Defining the wave function of the universe with fractal metrics.

Conclusion

Fractal wave functions offer a new mathematical framework at the intersection of quantum mechanics + fractal statistics. Especially fractional derivatives, the Hurst exponent, and multifractal distributions are the cornerstones of this model.

Fractal Wave Functions – Example Calculations

Below are example calculations regarding the mathematical use of fractal wave functions. These examples show the expansion of the classical quantum mechanics wave function with fractal parameters.

1. Fractal Amplitude Calculation

Fractal amplitude function:

𝐴(𝑥) =∣ 𝑥 ∣^𝐻

Example: For 𝐻 = 0.7, 𝑥 = 5:

𝐴(5) =∣ 5 ∣^0.7 ≈ 3.08

This indicates that the amplitude of the particle is scale-dependent.

2. Fractal Phase Calculation

Fractal phase function:

𝜙(𝑥, 𝑡) = 𝛼 ⋅ 𝑥^𝐷_f + 𝛽 ⋅ 𝑡^𝐷_f

Example: 𝛼 = 0.5, 𝛽 = 0.2, 𝐷f = 1.3, 𝑥 = 4, 𝑡 = 2:

𝜙(4, 2) = 0.5 ⋅ 4^1.3 + 0.2 ⋅ 2^1.3 ≈ 3.65

3. Fractal Wave Function

Fractal wave function:

Ψ(𝑥, 𝑡) = 𝐴(𝑥, 𝑡) ⋅ 𝑒 ^{i⋅𝜙 (𝑥,𝑡)}

Example: 𝐴(5) = 3.08, 𝜙(4, 2) = 3.65:

Ψ(5, 2) = 3.08 ⋅ 𝑒^{i ⋅ 3.65}

If we expand it using Euler’s formula:

Ψ(5, 2) = 3.08 ⋅ cos(3.65) + 𝑖 ⋅ sin(3.65)

Ψ(5, 2) ≈ −2.83 + 𝑖 ⋅ −0.92

4. Simple Solution for Fractal Schrödinger Equation

Fractal Schrödinger equation:

𝑖ℏ (𝜕^𝛼/𝜕𝑡^𝛼) Ψ(𝑥, 𝑡) = − (ℏ²/2𝑚)∇^2𝐷_f Ψ(𝑥, 𝑡) + 𝑉(𝑥)Ψ(𝑥, 𝑡)

Example parameters: 𝛼 = 0.8, 𝐷_f = 1.2. In this case, the derivatives are calculated with fractional derivative operators, not classical linear ones.

Conclusion

These examples show how fractal wave functions are calculated via the Hurst exponent, fractal dimension, and fractional derivatives. Instead of the deterministic structure of the classical wave function, a scale-dependent and multifractal structure emerges.

Fractal Wave Functions Graphical Representations

With graphical representations, the amplitude, phase, and wave structure in the complex plane of fractal wave functions can now be examined visually.

Thanks to these graphs:

• Fractal amplitude function → Shows the scale-dependent amplitude change depending on the Hurst exponent.
• Fractal phase function → Reveals the fractal phase surface with space-time parameters.
• Fractal wave function → Visualizes the combination of amplitude and phase in a spiral structure in the complex plane.

These visual representations make the theoretical formulas more concrete, clearly revealing the multiscale nature of fractal wave functions.