Framework for Understanding the Constants and Unit Conversions

 J. Rogers, SE Ohio, 08 Mar 2025, 1502

The idea is that many of the fundamental constants, such as h (Planck's constant), k (Boltzmann constant), and c (speed of light), can be viewed as unit conversions between different physical quantities. If h is simply a unit conversion, then we should be able to isolate conversion factors between Hz (frequency), kg (mass), and E (energy). Here's the reasoning:

1. Mass-Energy Relation:

   • We know that kg_E = c².
   • If we cancel out kg_E from h, we are left with the conversion factor that links Hz and kg. This conversion factor is represented as Hz_kg, and it has units of kg·s.

   Thus, we have:

   $$\text{Hz}_\text{kg} = \frac{h}{\text{kg}_E}$$
2. This factor, Hz_kg, is the unit conversion that, along with kg_E, defines h.

   And we know that Hz_kg does not depend on c, because there are no units of meter in Hz_kg. c^2 is just canceling out the c^2 that is inside h isolating the remaining Hz_kg factor.

3. Mass and Frequency Conversion:

   • We can express mass in terms of frequency
     
     
   $$\text{Mass} = \text{Frequency} \times \text{Hz}_\text{kg}$$

   This relationship works because the units of Hz_kg align the units of frequency with mass.

   Therefore, the definition of h is:

   $$h = \text{Hz}_\text{kg} \times \text{kg}_E$$

4. Scaling of Natural Units:

   • Hz_kg is the specific scaling factor needed to align mass and frequency with natural units when h = 1. Similarly, c would be the exact scaling of the meter that would make h = 1 and c = 1. These are the scalings of natural units.

5. Inclusion of Temperature Conversion:

   • We can hypothesize that k (the Boltzmann constant) would also include a K_Hz conversion factor, in addition to the Hz_kg and kg_E conversion factors that h contains.

   The conversion factor K_Hz can be expressed as:

   $$\text{K}_\text{Hz} = \frac{k}{\text{Hz}_\text{kg} \times \text{kg}_E}$$
6. K_Hz has units of Hz/K and is the conversion factor needed to convert temperature to frequency. Therefore, we can express temperature in terms of frequency as:$$\text{Frequency} = \frac{\text{Temperature}}{K_\text{Hz}}$$

   The definition of k is:

   $$k = K_\text{Hz} \times \text{Hz}_\text{kg} \times \text{kg}_E$$

7. Modular Framework:

   • These unit conversion factors (like Hz_kg, K_Hz, and kg_E) serve as simple modular units that scale temperature, frequency, mass, and energy. Just like in good programming practice, each factor has a clear purpose, making the relationships between these physical quantities simpler and easier to understand.

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Applying This Framework to Planck's Law

Let's now apply this framework to Planck’s law:

The traditional formula for Planck's law is:

$$B(f,T) = \frac{2hf^3}{c^2} \left( \frac{1}{e^{\frac{hf}{kT}} - 1} \right)$$

This formula is filled with constants and is not immediately clear in terms of its physical meaning. However, in our new framework, we can rewrite it more transparently as:

$$B(f,T) = \frac{\text{frequency energy}}{\text{frequency distribution factor}}$$

Where:

• frequency energy is:

  $$\text{frequency energy} = 2 f^3 \times \text{Hz}_\text{kg}$$

• frequency distribution factor is:

  $$\text{frequency distribution factor} = e^{\frac{f}{T K_\text{Hz}}} - 1$$
  

This formulation highlights how each term relates to a specific conversion factor.

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Understanding the Exponent Term

The exponential term in the denominator can be understood in multiple ways. Specifically:

$$e^{\frac{f}{T K_\text{Hz}}} - 1$$

This exponent can be seen in three ways:

• As hf / (kT) (energy-to-energy ratio),
• As f / (T K_Hz) (frequency-to-frequency ratio),
• Or as (f / K_Hz) / T (temperature-to-temperature ratio).

These are all algebraically identical forms, but they offer different perspectives on the relationship between energy, frequency, and temperature. You can change viewpoints just looking at the formula as written, this means the observer decides if this is a frequency, temperature, or energy ratio. This flexibility allows us to see the equation from the perspective of energy, frequency, or temperature, simultaneously.

This idea—of the same formula providing different perspectives depending on the observer’s focus—is a quantum understanding embedded in the equation itself. The observer can choose to see it in terms of energy, frequency, or temperature without altering the formula.

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Conclusion

This framework not only provides a cleaner understanding of the relationships between temperature, frequency, mass, and energy, but it also simplifies the interpretation of fundamental constants. By treating h, k, and c as unit conversions, we can better understand how they relate to each other and how they scale in natural units. Each constant has a clear modular role, making the underlying physics more transparent and accessible.