## Background

The energy of a 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 proportionality 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 complex units of kg m^{2}/s, which demonstrates that it is not truly a fundamental constant. Similar to the gravitational constant (G) and the Coulomb constant (k), the Planck constant is a *proportionality *constant that represents a combination of multiple constants.

**Photon creation**– when an electron moves to a lower energy level in the atom (closer to the nucleus), it emits a photon.**Photon absorption –**when an electron absorbs a photon it moves to a higher energy level in the atom or may leave the atom.

A detailed description of photons and how they are generated and adhere to the conservation of energy is provided in the section on photon creation and absorption.

## Explanation

In **energy wave theory**, photon energy is derived from the Energy Wave Equation, without need for the Planck constant. It is simply a transfer of longitudinal wave energy (E_{l}) to transverse wave energy (E_{t}). Since energy is always conserved, the relationship can be described using the same wave constants used for longitudinal waves.

The creation of a transverse wave was described in the photon wavelength page and is not recreated here. A photon can be expressed either in wavelength terms or in energy. Here in this section, the description of how energy is converted from one wave form to the next is described in detail.

**1) Orbital **(Electron + Proton)

An electron is attracted to a proton in the nucleus of an atom but it does not reach the proton with the rare exception being the electron capture process. The electron is attracted to the proton due to destructive longitudinal waves and moves to the point of minimal amplitude. It is also repelled by the proton with a strong, axial wave that decreases at the cube of the distance, creating a “gap” where attractive and repelling forces are zero. This is an orbital and explained in detail in the Atoms section.

If the electron is captured by an atom or moves to an orbital closer to the nucleus, it increases its destructive longitudinal wave energy with the proton. When in motion, it has kinetic energy. When it reaches the orbital, where there is minimal amplitude, it stops. It vibrates when coming to a stop to release the energy. This conservation of energy is expressed below in the animation.

There are three variables in the Transverse Energy Equation. One is the amplitude factor (δ) which is a convenient measurement of the constructive wave interference between the electron and proton. For a single electron and a single proton, this factor is one. Other configurations are found in the Amplitude Factor table. The other two variables are the initial position (r_{0}) of the electron relative to a particle it is interacting with and the final position (r). Orbital distances are found in the atomic orbitals calculation page.

**2) Annihilation **(Electron + Positron)

Unlike a proton, there is no repelling force in the positron. There is only an attractive force due to destructive longitudinal waves. Therefore, the electron does not have an orbital like it does with a proton. Instead, it reaches the point of minimal wave amplitude which is within the standing wave structure at nodes. At this position, an electron and positron annihilate and have complete destructive wave interference. The wave centers for each particle are still there, reflecting longitudinal waves, but they are completely destructive. Standing waves cannot form due to this destructive interference, which means the particles have no mass and are not detected.

The destruction of longitudinal wave energy is transferred to transverse energy. In the transfer, the entire particle’s standing waves (stored energy) is converted when each particle moves into final position and vibrates before coming to rest. This vibration creates two photons traveling in opposite directions as a result of both particles vibrating.

**3) Other Photon Interactions**

There are a handful of other scenarios of photon interactions with electrons as they are created or absorbed, but in all cases, energy is conserved. Longitudinal energy, transverse energy and kinetic energy (KE) can always be calculated. A summary of these other interactions was presented in the photon creation and absorption page and the complete description of photon experiments and the conservation of energy for each scenario is found in the *Photons *paper.

## Proof

Proof of the energy wave explanation for photon energies:

- Calculations of 250+ photon energies for ionization of hydrogen to calcium (neutral and ionized elements).
- Calculations of the conversion of energy from particles to photons for annihilation, orbital transitions and ionization.
- Derivation of the Planck constant (h).
- Derivation of Planck’s relation (E=hf).

**Hydrogen Ground State Energy (**Rydberg Energy)

The calculation of the Rydberg energy is shown with the Transverse Energy Equation in **two formats** (classical constants and wave constants). Both result in the same solution.

**Variables:**

- δ
_{H}**=**1 - r
_{0}= infinity (*electron captured from outside atom*) - r = a
_{0}= 5.2918E-11 m (*Bohr radius for ground state hydrogen)*

**Equation #1 : **Transverse Energy Equation – Classical Format

**Result: **2.1799E-18 joules (kg m^{2}/s^{2})

**Equation #2: **Transverse Energy Equation – Wave Format

**Result: **2.1799E-18 joules (kg m^{2}/s^{2})

**Comments:** 0.000% difference from the CODATA value for the Rydberg unit of Energy in both formats. Photon energies for the first twenty elements, from hydrogen to calcium, are accurately calculated using these equations.