Wavelength - Illuminating Science

The wavelength of an elementary particle with or without rest mass is calculated by applying the wavelength operator to the maximum Planck length potential.

Wavelength formula solves for a particle’s wavelength; wavelength fits into the template as…

Standard formula

The standard wavelength formula applies the wavelength operator to the Planck length potential

$\lambdabar = \displaystyle\frac{1}{\beta_{\lambda}} l_P$

The length unit dimension quantifies the particle’s wavelength as shown in the particle mechanics template.

Since length potential is the inverse of length units, the formula either multiplies the Planck length by the inverse operator, or divides the Planck length by the standard operator

Applying the wavelength operator to the maximum length potential in unit dimension L produces the observable quantity of wavelength.

Potential

Operator

Observable

The standard form of the wavelength formula uses the three equivalent forms of the inverse wavelength operator multiplied by the Planck length

Expanded formula

The formula can also use the expanded form of the wavelength operator to find the wavelength. The expanded form of the wavelength operator is the product of the rest mass and momentum operators

$\beta_{\lambda} = \beta_m \ \beta_p$ .

Substituting the two expanded operators produces the same transformation to the particle’s wavelength.

The two operators produce the same wavelength observable from the Planck length potential.

Potential

Operators

Rest mass operator, ratio of Planck length to Compton wavelength — β_m

Rest mass operator, ratio of Compton wavelength to de Broglie wavelength — β_p

Observable

The expanded form of the wavelength formula applies the two operators to the Planck length potential

$\lambdabar = \displaystyle\frac{l_P}{\beta_m \ \beta_p}$

The expanded form of the wavelength formula uses three equivalent forms of the inverse rest mass and momentum operators multiplied by the Planck length.

de Broglie wavelength

Because the momentum and velocity operators are equal, additional forms of the wavelength formula can be created by substituting the velocity operator for the momentum operator

$p = \beta_m \ \beta_v p_P$

This is the familiar form of the equation known as the de Broglie wavelength formula. The equation produces the right answer but gives a poor representation of the particle.

The de Broglie wavelength formula can be stated using each of the three forms of the rest mass operator

Although velocity is the wrong description of a particle’s energy potential, the use of velocity to calculate the potential is pragmatic because mass and velocity are simple observables whereas the wavelength of a particle with rest mass is difficult to measure. This should not confuse the physical meaning of these operators however, which are explained under the rest mass operator.

The following illustration shows how the traditional formula for de Broglie wavelength using Planck’s constant is equal to the natural formula stated in fundamental Planck units

de Broglie wavelength: conversion from historical to natural formula

Compton wavelength

The Compton wavelength is a special case of the de Broglie formula and the New Foundation Model elementary wavelength formula when a particle’s velocity is equal to the speed of light. The formula applies the rest mass operator and the maximum value of the velocity operator to the maximum length potential.

Potential

Operators

Dimensionless proportionality operator, ratio of speed of light to speed of light — β_v

Observable

The Compton wavelength formula can be stated in each of the three fundamental unit dimensions. The following general form of the wavelength formula shows the Compton wavelength as a special case where the particle’s velocity is the speed of light

The simplified form of the Compton wavelength formula omits offsetting quantities of the speed of light

The following illustration shows how the traditional formula for Compton wavelength using Planck’s constant is equal to the natural formula stated in fundamental Planck units

Compton wavelength: conversion from historical to natural formula