## Background

A description of photons and how they are generated was provided in the section on Photon Wavelengths. The energy of the photon is related to its frequency/wavelength and calculated using the Planck relation, E=hf where energy (E) is related to the frequency (f) and a constant known as the Planck constant (h). Note that frequency and wavelength are related as frequency is the speed of the wave divided by wavelength.

Although photon energy is related to its frequency, the Planck constant is a convenient value in the E=hf equation. It resolves the energy equation correctly, but the Planck constant itself has units of kg m^{2}/s, which demonstrates that it is not truly a fundamental constant.

## Photon Energy

In energy wave theory, photon energy is also derived from the base energy equation, without need for the Planck constant. For reference, the base equation is in the form:

Particles generate a longitudinal, traveling wave. When vibrating, they will create a secondary, transverse wave perpendicular to the direction of motion. This creates two waves in two directions – or two photons. In the figure below, a particle is shown to create two photons with a transverse wave along with its longitudinal, traveling wave. The energy equation for this wave will have both a transverse frequency and longitudinal frequency where V_{t} is the volume of the cylindrical photon. It can also be thought of as the electrical and magnetic components of the electromagnetic wave.

The origin of this equation is the energy wave equation, substituting the aforementioned volume, frequencies and amplitudes. Note that the traveling wave is no longer spherical and the longitudinal amplitude is modified with the volume change. It is A_{lt}^{2} and its amplitude is reduced proportional to each wavelength. Its value is not known, nor is the amplitude for the transverse wave (A_{t}), thus the equation must be further derived to arrive at the Transverse Energy Equation.

The complete derivation is available in the *Particle Energy and Interaction* paper, although a partial derivation is found in the section on E=hf. During the derivation, the Planck constant is apparent. Not only is the equation derived from the fundamental energy wave equation, but furthermore, the Planck constant is calculated accurately as one of the fundamental physical constants derived by wave constants.

Photon energies are calculated using the difference in energy as particles annihilate or electrons change orbits, so the above equation is represented as a change in energy difference in the following equations. The radius r_{0} is the initial position of the particle relative to a particle it is interacting with and r is the final position.

The final derivation of the Transverse Energy Equation is:

**Transverse Energy Equation**