J. Rogers, SE Ohio, 22 Jan 2025, 1149
Abstract:
Planck's constant (h) is a fundamental constant in quantum mechanics, often regarded as an indivisible quantum of action. This paper proposes a reinterpretation of Planck's constant, suggesting that it is not a fundamental entity in itself, but rather an emergent property derived from the equivalent mass of a photon at 1 Hz and the square of the speed of light (c²). By carefully exploring the relationships between mass, energy, momentum, and frequency, we aim to show how h arises from the interaction of mass and spacetime, offering a unified perspective that bridges classical and quantum mechanics.
1. Introduction:
Planck's constant (h) is a ubiquitous constant in quantum mechanics. It relates the energy of a photon to its frequency (E=hf) and appears in various quantum equations. It is often treated as a fundamental constant of nature, but this paper will explore the hypothesis that perhaps it is not fundamental, and that it can be derived. The goal of this paper is to provide a step by step explanation of this idea.
2. The Photon at 1 Hz: Setting the Stage
Let's begin by considering a photon with a frequency of 1 Hz. This is a very low-frequency photon, and it is a useful reference point for us to define the other values that we want to discuss. For this photon, we have:
Frequency (f) = 1 Hz
Wavelength (λ) = c (since λ = c / f)
Energy (E) = h (since E=hf)
Momentum (p) = h/c
Equivalent Mass (m) = h/c^2 (from E=mc^2)
3. Mass, Energy, and the Speed of Light
One of the most famous equations in physics is Einstein's mass-energy equivalence: E = mc². This equation implies that energy and mass are interchangeable. The equivalent mass, calculated from the photon's energy, can be used to determine its contribution to the curvature of spacetime. It's important to clarify that while photons do not have rest mass, they do have an equivalent mass derived from their energy, and that is what we are referring to when we talk about the "mass" of a photon.
4. Re-examining Planck's Constant
If we calculate the equivalent mass of a photon at 1Hz, and multiply that by the speed of light squared, we arrive at the energy of the photon which is also defined as h. We find that:
h = (mass of a 1Hz photon) * c²
This is a critical observation, and the main point of this paper. Planck's constant is not some fundamental number; it is a consequence of the mass of a photon at 1Hz. Rather than Planck's constant dictating energy, rather the energy of a photon is simply the mass at 1Hz multiplied by c^2. It's the relationship that is fundamental, rather than the values themselves.
5. Momentum as a Function of Mass
Using the equivalent mass of the photon at 1Hz we can derive the momentum of that photon using the classical definition of momentum: p = mv. And indeed, we find that the momentum of the photon is derived from this mass.
6. The Worldline and Frequency
The equivalent mass of the photon at 1Hz is an invariant, but when the photon's frequency is increased, the momentum and energy increase along with it. The relationship between frequency, mass and spacetime can be thought of as the photon's worldline, and this worldline interacts with space-time, and the interaction we observe through mass, energy and momentum. It is important to emphasize that frequency is not the cause of this, but is rather just another way of observing this fundamental interaction.
7. Scale Invariance and Units
It is also important to note that the numerical values of these constants, such as h, are dependent on the units we choose. If we change the scale of our units of length, we also change the numerical value of Planck's constant. However, the relationship of h = (mass of a photon at 1 Hz) * c² always holds true. The mass of a photon remains invariant, but h scales as we choose our units.
8. Reinterpreting E=hf
We can re-write the formula for the energy of the photon as E = (m * f) * c² where 'm' is the equivalent mass of a photon at 1 Hz. This shows that the formula for energy (E=hf) has always been a formula for the interaction of the photon's mass with spacetime.
9. Implications and Further Exploration
This new understanding of Planck's constant has deep implications for quantum mechanics. If h is an emergent constant, then mass and spacetime may be more fundamental. This means that mass is a more fundamental quantity than we normally consider it to be, and that mass and gravity may be the fundamental forces that give rise to all other effects that we observe. Further research is needed to explore the mathematical implications, and to test if this framework can provide new insights or predictions.
10. Validation through Numerical Consistency
To validate the relationship h = (mass of a 1 Hz photon) ⋅ c², consider the equivalent mass of a photon at 1 Hz. This mass is approximately 7.3724973 × 10⁻⁵¹ kg. Using Einstein's equation E = mc², we find:
E=(7.3724973×10⁻⁵¹kg)⋅(299792458 m/s)²≈6.62607015×10⁻³⁴ J
This result is precisely the accepted value of Planck’s constant, h, at 1 Hz. This agreement shows that h can be derived from the effective mass of a photon at 1 Hz and the speed of light squared, using only established physical principles. This strongly suggests that Planck's constant is a result of the equivalent mass at 1Hz, and does not need to be considered a fundamental constant in its own right.
Furthermore, this relationship demonstrates that h is not a standalone fundamental constant but a direct consequence of the effective mass of photons, which scales linearly with frequency. The photon's worldline in curved spacetime governs this scaling, encoding the interaction between mass, frequency, and spacetime curvature.
By reinterpreting
E = hf
as
E = (m⋅ f) ⋅ c²,
where m is the equivalent mass of a photon at 1 Hz, we see that h encapsulates this mass-energy equivalence in a compact and elegant form. The value of h arises naturally as the product of the effective mass at 1 Hz and c², and this mass scales by the frequency, changing the momentum and energy that are derived from that motion of curved space time in space time, unifying classical and quantum perspectives.
Furthermore, we can think of E=mc² as a modified kinetic energy formula, where v=c. This suggests that the formula for energy is simply the consequence of the interaction of mass and spacetime at the speed of light, and is not something mysterious.
11. Conclusion
We have shown that Planck's constant is a value that is derived from the equivalent mass of a photon and it's interaction with space time. To all effects h encodes this mass times c squared. By using this framework, we can see that classical mechanics and quantum mechanics are not separate concepts, but rather are manifestations of the interaction of mass and spacetime, that energy and momentum are derived from that interaction, and that mass is the most fundamental of the concepts we have explored today.
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