## Background

The photon is the carrier of the electromagnetic wave, which is responsible for light, radio waves, microwaves, X-rays, etc. All of these types of waves are based on the same electromagnetic wave but are differentiated by their wavelengths. Photon wavelength has an inverse relationship to photon energy. For example:

- A gamma ray is a photon with very high energy, but a very short wavelength
- A radio wave is a photon with low energy, but a long wavelength

## Explanation

In energy wave theory, a transverse wave is created from a particle that is vibrating perpendicular to the direction of wave motion. A faster vibrating particle results in a transverse wave with a shorter wavelength than a particle that vibrates slower. The greater the longitudinal amplitude differences in a particle’s interaction with surrounding particles, the faster the particle’s vibration. In an atom, one photon will escape and the other photon will be absorbed by the nucleus and cause it to recoil.

The creation of photons from electrons are not only seen within the atom, but also in the annihilation of an electron and its antimatter counterpart, the positron. In this case, two photons with short wavelengths (gammay rays) are created during the annihilation.

**Transverse Frequency & Wavelength**

In addition to the particle’s longitudinal wave, the vibration creates a secondary, transverse wave that includes a new transverse amplitude and wavelength component. There is now a longitudinal component and a transverse component in the photon. The differences in the types of waves in the electromagnetic spectrum, however, are dependent only on the wavelength of the transverse component. An example of a spherical electron vibrating to create the transverse wave is shown below.

The transverse frequency and wavelength is based on the speed of the vibration of the particle, which itself is based on the longitudinal energy difference from the electron’s starting position and ending position relative to the atomic nucleus. It is calculated using either the transverse frequency or wavelength equations (frequency is wave speed divided by wavelength). These equations can be used for any atom using the Photon Frequency or Photon Wavelength equations. Photons beyond hydrogen are often reported in terms of energy, not frequency or wavelength.

The equation requires a starting position (r_{0}) and ending position (r) of the electron. It also depends on constructive wave interference, which is given a variable called amplitude factor (δ) to simplify calculations of interference from many particles. This is explained in a visual below. The calculations for the electron’s position and amplitude factor are found in tables under atom calculations*.*

The complete derivation is found in *Particle Energy and Interaction, *but the final result was described on the photon equation page. The photon wavelengths for hydrogen are accurately calculated using this equation.

## Proof

Proof of the energy wave explanation for photon wavelengths:

- Calculations of hydrogen photon wavelengths for ionization and orbital transitions (
*see an example below*) - Derivation of the electron Compton wavelength

**Hydrogen Wavelength (1s Ionization****)**

In hydrogen photon wavelength calculations, the wavelengths of absorbed photons for hydrogen were calculated at differing orbitals when the atom is ionized. The 1s ionization wavelength is shown with the photon wavelength equation in **two format**s (classical constants and wave constants). Both result in the same solution.

In the case of ionization, this equation can be simplified. The electron is ejected from the atom so the final position (r) can be replaced by infinity in the equation. The 1s orbital uses the Bohr radius (a_{0}) as the distance – 5.2918E-11 m. The amplitude factor (δ) for hydrogen is one. The table of amplitude factors is found in atom calculations.

**Variables:**

- δ
_{H}**=**1 - r
_{0}= a_{0}= 5.2918E-11 meters (m) - r = infinity (
*leaves the atom in ionization*)

**Equation #1: **Photon Wavelength Equation – Classical Format

**Result: **-9.113E-8 meters (m)

**Equation #2: **Photon Wavelength Equation – Wave Format

**Result: **-9.113E-8 meters (m)

**Comments:** Both formats result in a 0.004% difference from measured results. A negative sign indicates a photon is absorbed by the atom; a positive sign is the creation of a photon.