# de Broglie's Equation Solver

## de Broglie's Equation Solver

This equation solver for de Broglie's Equation is a proof of concept for the notion of parametric recursive forms (RFs). The velocity of a particle must be set as a parameter.
Solve de Broglie's Equation

### Background

This tool, in the modality of an equation solver for de Broglie Equation, is a proof of concept for the notion of parametric recursive forms (RFs). To access the RF you first need to set the velocity of a particle as a parameter.

RFs automatically submit themselves when a user completes the following steps (see Theory section).

1. Select an initial option or accept the default one.
2. Input a value compatible with said option.
3. Select a new option.

### Examples

The following examples illustrate how this RF works. Note that data in scientific notation must be entered in "e" notation, with the + sign being optional, but not the − one.

A user wants to calculate the de Broglie wavelength of a particle moving at a velocity of 1.66 x 106 m/s. To do this, he enters 1.66e6 in the RF parameter field and presses the "Set" button.

Once he accesses the RF, he selects the mass option and enters 2.33e-33. When he switches to the wavelength option, the RF returns 1.71e-7 as the de Broglie wavelength, in meter. Done!

If he now switches to the frequency option, said output becomes the new input and the RF returns 9.71e+12 as the de Broglie frequency, per second (hertz).

Switching to the momentum option, the previous output becomes the new input so the RF returns 3.88e-27 as the de Broglie momentum, in kilogram·meter per second.

Finally switching to the energy option, the previous output becomes the new input and the RF returns 6.44e-21 as the de Broglie energy, in joules.

### Discussion

We define a recursive form (RF) as an HTML form where the name/value pairs of successful controls are recursively generated through the decision-making process of users.

### Algorithm

The algorithm implemented by this RF represents all possible missing terms of de Broglie Equation as a two-dimensional associative array of the form

\$index = [
[
"missing_term_1" =>"new_option_1",
"solution_1_1" => "value_1_1",
"solution_1_2" => "value_1_2",
⋮
],
[
"missing_term_2" =>"new_option_2",
"solution_2_1" => "value_2_1",
"solution_2_2" => "value_2_2",
⋮
],
⋮
[
⋮
...
]
];


This is faster than using nested switch statements, particularly as the size or dimensionality of the array increases.

The index is searched in two steps. First, the subarray that matches the missing term (i.e., the new option) is located. Subarray solutions are then located. If there is a match (i.e., in function of the initial option and input value), the solution is retrieved and displayed in the input field as the new value.

Users can repeat this process as many times as they wish to by simply switching to a new option.

### Derivation of de Broglie Equation

Prince Louis-Victor de Broglie proposed that light and matter exhibit wave-like and particle-like properties. Thus for a moving particle of mass $$m$$, velocity $$v$$, and momentum $$p = mv$$, we may compute its de Broglie's wavelength $$\lambda$$ by combining Einstein's and Planck's Equations, and replacing the speed of light $$c = \lambda\nu$$ with the velocity $$v$$ of the particle.

(1)$E = mc^2 = h\nu$

(2)$E = mv^2 = h\nu$

(3)$E = pv = h{v\over \lambda}$

(4)$p = {h\over \lambda}$

(5)$\lambda = {h\over p}$

where Planck's constant is $$h = 6.63 \times 10^{-34} J \cdot s$$ and $$\nu$$ is the particle's frequency. Some authors use the letter f for frequency to prevent readers from mistaking the greek letter nu, $$\nu$$, for velocity.

Solving (1) for $$m$$, we get $$m = {h\nu\over c^2}$$ so we may imputate an effective mass to a photon. As noted by Dr. Sten Odenwald at the Astronomy Cafe (Odenwald, 2021):

"This equation says that the energy carried by a photon which has NO REST MASS, is equivalent to an amount of ordinary mass in grams, and that this 'effective mass' varies with the frequency of the photon."

This parametric RF was designed to solve the above equations for a missing term.

### Exercises

• Calculate the de Broglie wavelength, frequency, momentum, and energy for an electron moving at 5.0e+6 m/s. The mass of an electron is 9.11e-31 kg.
• A proton moving at 2.83 x 104 m/s has momentum of 4.73e-23 kg·m/s. Compute its energy, mass, wavelength, and frequency.