Reinterpreting Planck's Constant: From Fundamental Constant to Geometric Unit Conversion Factor

James Rogers, SE Ohio, 13 Nov 2024 Noon

Abstract

This paper presents a novel analysis of Planck's constant ($h$) and its role in quantum mechanics. We argue that $h$ is not a fundamental constant of nature, but rather a unit conversion factor arising from our chosen measurement units divided by the speed of light. Through unit definition changes, we demonstrate how Planck’s constant serves as an encoded form of the classical wave equation embedded with unit scaling and inverse speed of light. This geometric reinterpretation of quantum mechanics challenges long-held assumptions, suggesting that quantum phenomena may be intrinsically geometric rather than probabilistic. This new perspective opens pathways for re-evaluating the foundations of modern physics.

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Introduction

For over a century, Planck's constant has been regarded as a fundamental constant in quantum mechanics. Max Planck introduced $h$ while solving the black body radiation problem, a discovery that eventually paved the way for the quantum revolution. However, our analysis proposes that $h$ is not fundamental; rather, it arises from our arbitrary choices of unit systems. We propose that $h$ is essentially a geometric scaling factor, obscuring the underlying simplicity of quantum mechanics. This paper presents a framework that demystifies $h$ and reveals quantum mechanics as grounded in fundamental geometric relationships.

Why Planck’s Constant Was Historically Regarded as Fundamental

Historically, Planck's constant has been viewed as a cornerstone of quantum mechanics, central to the equations governing energy quantization, wave-particle duality, and atomic structure. Max Planck introduced $h$ to resolve the ultraviolet catastrophe, and its adoption enabled Einstein to quantify light as photons, leading to the photoelectric effect. Since then, $h$ has been embedded in numerous quantum equations, solidifying its status as a bridge between energy and frequency. Yet, this perspective may have limited our understanding, framing quantum mechanics through an arbitrary constant rather than questioning if $h$ is simply a byproduct of human-defined units divided by the speed of light,

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Methodology

To test the hypothesis that $h$ is a unit conversion factor divided by c, we developed a Python program to manipulate the base units of length (m), mass (kg), and time (s) and observe the effect on $h$ and the speed of light ($c$). Specifically, we sought to preserve the product $hc$, a value consistent with our target scaling factor $K$, which we define as:

$$K=hc$$

We created three functions to adjust $h$ and $c$ by redefining units in terms of meters, kilograms, and seconds. Each function recalibrates $h$ by modifying one of these base units, preserving the product $hc$ and demonstrating how $h$ functions as a unit scaling factor rather than a fundamental constant.

Methodology for Scaling Constants $h$ and $c$

The methodology begins by recognizing that the product of Planck's constant ($h$) and the speed of light ($c$) has units of $\text{kg} \cdot \text{m}^3 / \text{s}^2$. To scale these constants, the ratio between the original product $h \cdot c$ and the target product is first calculated. This ratio determines the scaling factor to apply. The scaling factor is then adjusted by the power of the unit being adjusted in the definition of $h$ or $c$, and then used to scale $h$ or $c$. By applying this ratio, both constants are scaled in a manner that maintains the integrity of the unit relationships in the new context, ensuring consistency across the transformation.

The following functions are from this project.
https://github.com/BuckRogers1965/RedefineUnitsForPlancksConstant

import numpy as np
current_hc = h * c

# Adjusts the unit of meters and calculates new values for h and c
def adjust_m_and_calculate(initial_h, initial_c, target_hc):
    meter_adjustment = np.cbrt(current_hc / target_hc)
    new_c = initial_c / meter_adjustment
    new_h = initial_h / meter_adjustment**2
    return meter_adjustment, new_h, new_c

# Adjusts the unit of kilograms and calculates new values for h
def adjust_kg_and_calculate(initial_h, initial_c, target_hc):
    kg_adjustment = current_hc / target_hc
    new_h = initial_h / kg_adjustment
    return kg_adjustment, new_h, initial_c

# Adjusts the unit of seconds and calculates new values for h and c
def adjust_s_and_calculate(initial_h, initial_c, target_hc):
    second_adjustment = np.sqrt(current_hc / target_hc)
    new_c = initial_c / second_adjustment
    new_h = initial_h / second_adjustment
    return second_adjustment, new_h, new_c

Results

The program demonstrates that by redefining the base units, we can achieve any desired value for $h\mathrm{c.\; Notably,\; setting }$$hc = 1 \, \text{J} \cdot \text{m}$ yielded:

• 1 new m = $5.835 \times 10^{-9}$ old m
• 1 new kg = $1.986 \times 10^{-25}$ old kg
• 1 new s = $4.457 \times 10^{-13}$ old s

In this recalibrated system, the relationship $E = hc/\lambda$ simplifies to $E = 1/\lambda$, showing that the essence of the formula is purely geometric. If we assume 1m wavelength for convenience, $1 \, \text{J}$ of energy corresponds to $1 \, \text{J} \cdot \text{m} / 1 \, \text{m}$, at these unit definition, indicating that $h$c is simply converting units for inverse wavelength to energy, obscuring the intrinsic geometry. And in each case h = 1/c at that unit definition. 

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Analysis: A Geometric Interpretation of Quantum Mechanics

This result suggests that the relationship $E = 1/\lambda$ is a geometric one, unencumbered by the unit-based scaling of $h$ and $c$. The traditional view, where $E = hf$ or $E = hc/\lambda$, imposes an implicit scaling that is not essential to the underlying physics.

The Role of Frequency and Wavelength

In this framework, $h$ effectively embeds a factor of $1/c$ that converts frequency to wavelength. By defining $K = hc$, we can rewrite $h = K/c$ and substitute directly into energy formulas:

$$E = hf \implies E = \frac{Kf}{c} = \frac{K}{\lambda}$$This interpretation reveals that 

$h$ was never a mystical constant but a convenient device for unit conversion, obscuring a purely geometric relationship between frequency, wavelength, and energy.  f/c is simply 1/λ from the formula c = fλ.

Implications for Wave-Particle Duality and the Probabilistic Nature of Quantum Mechanics

Reinterpreting $h$ as a unit conversion factor reshapes our understanding of quantum phenomena. It suggests that the wave-particle duality and probabilistic nature of quantum mechanics arise from our unit choices rather than any intrinsic randomness or duality in nature. By simplifying to $E = K/\lambda$, we see that quantum mechanics could be fundamentally geometric, with probabilistic interpretations merely artifacts of unit manipulation.

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Practical and Experimental Implications of Recalibrated Units

Though impractical for daily use, these redefined units reveal $h$ as simply a unit scaling factor divided by $c$. Experimental settings traditionally rely on $h$ to maintain consistency with established measurement units. However, adopting recalibrated units as a thought experiment could provide fresh insights, allowing us to reframe equations like $E = K/\lambda$ without traditional constants, potentially simplifying experimental analysis at high energies and frequencies. We may have been missing many simple geometric relationships because of our unit scaling.

Further Research Directions

The revised framework invites further exploration into rewriting quantum mechanical functions and constants. Replacing $h$ with $K/c$ and $\hbar$ with $K/(2 \pi c)$ in quantum equations could reveal additional geometric relationships, refining our understanding of wave-particle interactions, atomic structures, and even large-scale cosmological phenomena.

Comparison to Existing Theories

Our reinterpretation of $h$ contrasts sharply with established quantum mechanics and raises fundamental questions about the nature of constants with units. If $h$ can be reduced to a unit scaling factor divided by speed of light, might other constants similarly serve as disguised unit conversions divided by another constant? Such inquiries could lay the groundwork for a unified theory that encompasses both classical and quantum mechanics, based entirely on geometric principles. If the same constant is alway being multiplied by the same factor in every equation it is being used in them

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Conclusion

This analysis reinterprets Planck's constant as a unit-derived factor rather than a fundamental constant. By redefining $h$ as a product of $K$ and $c$, we see that quantum mechanics is not fundamentally probabilistic but geometric. Our findings challenge traditional quantum mechanics, suggesting that it has been obscured by human-defined units, which mask a simpler geometric truth. We conclude that quantum mechanics, and potentially all physics, may be grounded in these relationships, urging a re-evaluation of foundational assumptions and inviting further exploration into the nature of all constants in physics.