Reinterpreting Planck's Constant: A Geometric Perspective on Quantum Mechanics

Each new post is an evolution in my thinking.  Think of this blog as my stream of conscious thoughts formalized. 

James Rogers, SE Ohio, 14 Nov 2024 1225 

Abstract

This paper explores the relationship between frequency, wavelength, and energy through the lens of Planck's constant ($h$). We propose that $h$ serves as a geometric unit conversion factor that projects frequency ($f$) into hyperbolic space via the division by the speed of light ($c$). The reciprocal relationship between frequency and wavelength leads to a return from hyperbolic wavelength space to linear energy space, scaled by the product $hc$. This reinterpretation challenges traditional views of quantum mechanics, suggesting that what has been perceived as quantization may instead arise from our unit scaling choices.

Introduction

For over a century, Planck's constant has been regarded as a fundamental aspect of quantum mechanics, central to the equations governing energy quantization. However, this paper argues that $h$ is not an intrinsic property of nature but rather a result of our arbitrary selection of measurement units. By examining how $h$ functions in relation to frequency and wavelength, we reveal a geometric framework that underlies quantum phenomena.

Theoretical Framework

Projection into Hyperbolic Space

In classical wave mechanics, the relationship between frequency ($f$) and wavelength ($\lambda$) is given by:

$$c=f\lambda$$

From this equation, we can express wavelength as:

$$\lambda =\frac{c}{f}$$

When we divide frequency by the speed of light ($c$), we effectively project it into hyperbolic space:

$$\frac{f}{c}=\text{inverse wavelength}=\frac{1}{\lambda }$$

This projection allows us to conceptualize how changes in frequency correspond to changes in wavelength in a hyperbolic manner.

Returning to Linear Energy Space

The energy associated with a photon is given by:

$$E=hf$$

Substituting our expression for frequency in terms of wavelength yields:

$$E=h\left(\frac{c}{\lambda }\right)=\frac{hc}{\lambda }$$

The 1/λ returns the wavelength to a linear energy space that is scaled by the ratio returned by hc that is the ratio between h and 1/c and that ratio is just unit scaling.

This demonstrates that energy can be expressed as inversely proportional to wavelength when scaled by $hc$. Thus, the apparent quantization observed in energy levels arises from this scaling factor rather than being an inherent property of energy itself.

The Role of $hc$

The product $hc$ serves as a unit scaling factor that transforms our understanding of energy. By recognizing $h$ as a geometric conversion factor rather than a fundamental constant, we can simplify our analysis:

1. Geometric Interpretation: The relationship between frequency, wavelength, and energy can be understood geometrically without relying on constants like $h$.
2. Unit Scaling: The scaling introduced by $hc$ obscures the underlying simplicity of these relationships.

Methodology

To illustrate these concepts, we developed a Python program that analyzes the relationship between frequency and energy while manipulating units. By redefining base units (meters for length, kilograms for mass, and seconds for time), we preserved the product $hc$ while observing how changes affect our understanding of quantum mechanics.

Results

The program demonstrated that redefining units could lead to simplified relationships in quantum mechanics. For instance, if we set $hc=1\text{ J}\cdot \text{m}$, we found that:

• Energy: Simplified to $E=\frac{1}{\lambda }$

This indicates that the essence of quantum mechanics may be purely geometric rather than dependent on arbitrary constants.

Discussion

Implications for Quantum Mechanics

Our findings suggest that traditional interpretations of quantum mechanics may have been limited by our reliance on constants like $h$. By viewing these constants as unit scaling factors rather than intrinsic properties of nature, we open new pathways for understanding wave-particle duality and the probabilistic nature of quantum phenomena.

1. Geometric Relationships: Quantum mechanics may be fundamentally geometric, with probabilistic interpretations arising from unit manipulation.
2. Reevaluation of Constants: If $h$ can be redefined in this manner, other physical constants may also serve as disguised unit conversions.

Future Research Directions

The framework established in this paper invites further exploration into rewriting quantum mechanical functions and constants. By replacing $h$ with $K\mathrm{/}c$ (where $K=hc$) in quantum equations, researchers could uncover additional geometric relationships crucial for understanding atomic structures and wave-particle interactions.

Conclusion

This analysis reinterprets Planck's constant as a unit-derived factor rather than a fundamental constant. By recognizing how it projects frequency into hyperbolic space and returns through reciprocal relationships back into linear energy space scaled by $hc$, we challenge traditional views of quantum mechanics. Our findings suggest that quantum phenomena may be grounded in simpler geometric truths obscured by human-defined units. This perspective urges a reevaluation of foundational assumptions in physics and invites further exploration into the nature of all physical constants.


Appendix A

This program demonstrates how this perceived quantization occurs step by step as outlined in the program. 

import numpy as np

import pandas as pd

# This program generates a table of frequencies, their corresponding inverse wavelengths (frequency / c),

# and their energy equivalents scaled by hc. Traditionally, the product hc and the appearance of discrete

# energy values have been interpreted as evidence of intrinsic quantization in atomic and subatomic processes.

#

# However, this calculation highlights an alternative interpretation: the apparent "quantization"  

# in energy results from the scaling factor hc applied in reciprocal (hyperbolic) space 

# of 1/wavelength rather than any intrinsic, indivisible energy steps. The hyperbolic increments in energy 

# arise due to the reciprocal relationship between frequency and wavelength (1/frequency), which projects 

# continuous frequencies into hyperbolic space in a manner that appears quantized when scaled by hc.

#

# Consequently, this perspective suggests that quantization may be an artifact of how energy values are

# represented and scaled within our unit system, rather than an inherent property of energy itself. This 

# program, therefore, offers a different lens for understanding quantum mechanics, where the appearance of 

# discrete energy levels is a result of unit scaling in hyperbolic space.

# Constants

c = 299792458       # Speed of light in m/s

h = 6.62607015e-34  # Planck's constant in J·s

hc = h * c

# Parameters

start_frequency = 1.0e12  # Starting frequency in Hz

step = 0.1e12             # Step size in Hz

num_steps = 10            # Number of steps (you can adjust this as needed)

# Create lists to store the values

frequencies = []

wavelengths = []

f_over_c_values = []

f_over_c_hc_values = []

hf_energies = []

# Generate the table data

for i in range(num_steps):

    frequency = start_frequency + i * step

    f_over_c = frequency / c

    w = c/frequency

    f_over_c_hc = f_over_c * hc

    hf_energy = h * frequency

    

    frequencies.append(frequency)

    wavelengths.append(w)

    f_over_c_values.append(f_over_c)

    f_over_c_hc_values.append(f_over_c_hc)

    hf_energies.append(hf_energy)

# Create a DataFrame for easier viewing

table = pd.DataFrame({

    'Frequency (Hz)': frequencies,

    'λ (m)': wavelengths,

    '1/λ': "1 / λ =",

    'f / c (1/m)': f_over_c_values,

    '(f / c) * K (J)': f_over_c_hc_values,

    '(E = hf (J))': hf_energies

})

# Print the table

print (f"K = ratio between h and c, taken as hc with units J m")

print(table)

# --------------------------------------------------------------------------------

# Results:

#K = ratio between h and c, taken as hc with units J m

#    Frequency (Hz)     λ (m)      1/λ  f / c (1/m)  (f / c) * K (J)  (E = hf (J))

#0    1.000000e+12  0.000300  1 / λ =  3335.640952     6.626070e-22  6.626070e-22

#1    1.100000e+12  0.000273  1 / λ =  3669.205047     7.288677e-22  7.288677e-22

#2    1.200000e+12  0.000250  1 / λ =  4002.769142     7.951284e-22  7.951284e-22

#3    1.300000e+12  0.000231  1 / λ =  4336.333238     8.613891e-22  8.613891e-22

#4    1.400000e+12  0.000214  1 / λ =  4669.897333     9.276498e-22  9.276498e-22

#5    1.500000e+12  0.000200  1 / λ =  5003.461428     9.939105e-22  9.939105e-22

#6    1.600000e+12  0.000187  1 / λ =  5337.025523     1.060171e-21  1.060171e-21

#7    1.700000e+12  0.000176  1 / λ =  5670.589618     1.126432e-21  1.126432e-21

#8    1.800000e+12  0.000167  1 / λ =  6004.153714     1.192693e-21  1.192693e-21

#9    1.900000e+12  0.000158  1 / λ =  6337.717809     1.258953e-21  1.258953e-21