DeBroglie Wavelength

The De Broglie Equation calculator computes the wavelength of a particle based on the Planck's Constant and momentum (p = m•v).

INSTRUCTIONS: Choose units and enter the following:

• (m) Mass of Particle
• (v) Velocity of Particle

Wavelength (λ): The calculator returns the DeBrogile wavelength in nanometers (nm).  However, this can be automatically converted to other length units (e.g. angstroms) via the pull-down menu.

The Math / Science

The De Broglie equation is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It was proposed by Louis de Broglie in 1924 and forms the basis of wave-particle duality, which states that every particle or quantum entity can exhibit both wave-like and particle-like properties.

The DeBrogile equation is:

   λ = h/(m·v)

where:

• λ is the DeBrogile wavelength
• h is Planck's Constant
• m particle mass
• v particle velocity

Note: m⋅v is momentum.  So it is common to see this same formula in the following form:

   λ = h/p

where:

• λ is the DeBrogile wavelength
• h is Planck's Constant
• p is the momentum

De Broglie combined Einstein's famous energy equation, E = mc², and Planck's equation, E = hv, to create this equation.  The DeBrogile equation uses Planck's Constant (h = 6.626 x 10^-34 m²*kg/s) to calculate the wavelength associated with an object relating to its momentum (p = mv).

Implications of the De Broglie Equation

Wave-Particle Duality:

According to the De Broglie hypothesis, particles such as electrons, protons, and even larger objects can exhibit wave-like properties. This means that they can be described by a wavelength, similar to light waves.

Quantum Mechanics Foundation:

The concept that particles have wavelengths is a cornerstone of quantum mechanics, leading to the development of quantum theories that describe the behavior of particles at the atomic and subatomic levels.

Electron Diffraction:

One of the key experimental verifications of the De Broglie hypothesis is electron diffraction. When electrons are fired at a crystal, they produce diffraction patterns similar to those produced by X-rays, indicating wave-like behavior.

Applications of the De Broglie Equation

Electron Microscopy:

The short De Broglie wavelength of electrons allows electron microscopes to achieve much higher resolution than light microscopes.

Quantum Mechanics and Chemistry:

The De Broglie wavelength concept is critical in understanding the behavior of electrons in atoms and molecules, leading to the development of quantum chemistry and the Schrödinger equation.

Particle Physics:

It plays a crucial role in particle physics, helping to explain the wave-like behavior of particles in high-energy physics experiments.

Example

Calculate the wavelength (in meters) of an electron traveling 1.24 x 10⁷ m/s. The mass of an electron is 9.11 x 10^-28 g. 

Define variables:

• h = Planck's constant (6.626 x 10^-34 m²*kg/s)
• m = 9.11 x 10^-28 g = 9.11 x 10^-31 kg
• v = 1.24 x 10⁷ m/s

Substitute values into the De Broglie Equation:

λ = h/mv

λ = (6.626 x 10^-34 m²*kg/s) / (9.11 x 10^-31 kg) (1.24 x 10⁷ m/s)

λ = 5.86 x 10^-11m = 0.0586 nm

Energy, Photon and Wavelength Calculators

• Quantum Energy (E=h⋅v): Computes radiant energy in the Planck-Einstein relationship (E = h•ν) based on Planck's constant and a frequency of radiation.
• Energy of a Photon (E=h⋅f): Computes the energy of a photon based on the frequency and Planck's Constant.
• Photon Energy from Wavelength (E=(h⋅c)/λ): Computes the energy of a photon based on Planck's constant (h), the speed of light (c) and the wavelength of the photon ( λ)
• Photon Wavelength form Energy (λ=(h⋅c)/E): Computes the wavelength of a photon based on Planck's constant (h), the speed of light (c) and the energy of the photon (E).
• DeBroglie Wavelength (λ=h/(m⋅v)): Computes the wavelength of a particle based on the Planck's Constant and momentum (p = m•v).
• Plank Constant (h):  Fundamental constant in physics that plays a central role in quantum mechanics which relates the energy of a photon to its frequency

Supplemental Material

ChemWiki (UCDavis) : De Broglie Equation (with example)

References

Whitten, et al. "Chemistry" 10th Edition. Pp. 144