De Broglie Wavelength Calculator — Matter Wave, Momentum & Particle Wavelength

What is the de Broglie Equation?

According to Louis de Broglie's wave-particle duality principle, every moving particle has a wavelength associated with it. This de Broglie wavelength calculator helps you find that wavelength using the famous equation:

λ = h / (m × v) = h / p

Where:

• λ — de Broglie wavelength
• h — Planck's constant = 6.62607015 × 10⁻³⁴ J·s
• m — mass of the particle (kg)
• v — velocity of the particle (m/s)
• p = m × v — momentum of the particle (kg·m/s)

The Units of Mass

In quantum mechanics, particle masses are often expressed in special units. This calculator supports both metric (SI) and American (imperial) measurement systems:

Metric / SI Mass Units

• kg — kilogram (SI base unit)
• g — gram = 10⁻³ kg
• mg — milligram = 10⁻⁶ kg
• µg — microgram = 10⁻⁹ kg
• m_e — electron rest mass = 9.10938370 × 10⁻³¹ kg
• m_p — proton mass = 1.67262 × 10⁻²⁷ kg
• u — atomic mass unit (Dalton) = 1.66054 × 10⁻²⁷ kg

American (Imperial) Mass Units

• lb — pound = 0.453592 kg
• oz — ounce = 0.0283495 kg
• slug — slug = 14.5939 kg (used in US engineering)

Velocity Units Supported

The calculator supports both metric and American velocity units:

• m/s — meters per second (SI)
• km/s — kilometers per second
• km/h — kilometers per hour
• mph — miles per hour (American)
• ft/s — feet per second (American)
• % of c — percentage of the speed of light (c = 299,792,458 m/s)

What is the de Broglie Wavelength of an Electron?

Let's find the de Broglie wavelength of an electron traveling at 1% of the speed of light — a classic example.

Step 1: Mass of an electron:

m = 9.10938 × 10⁻³¹ kg

Step 2: Velocity at 1% of c:

v = 0.01 × 299,792,458 = 2,997,924.58 m/s

Step 3: Momentum:

p = m × v = 9.10938 × 10⁻³¹ × 2,997,924.58 = 2.7309 × 10⁻²⁴ kg·m/s

Step 4: de Broglie wavelength:

λ = h / p = 6.626 × 10⁻³⁴ / 2.7309 × 10⁻²⁴ = 2.426 × 10⁻¹⁰ m = 0.2426 nm

This wavelength falls in the X-ray range of the electromagnetic spectrum — comparable to the spacing between atoms in a crystal lattice, which is why electron diffraction is possible!

How to Calculate the de Broglie Wavelength of a Photon?

A photon has zero rest mass, but it does carry momentum. You can calculate its de Broglie wavelength using the momentum mode of this calculator.

Example: A photon with momentum p = 6.8 × 10⁻³⁵ kg·m/s.

λ = h / p = 6.626 × 10⁻³⁴ / 6.8 × 10⁻³⁵ = 9.74 m

This extremely long wavelength falls in the radio wave range.

Spectrum Regions by Wavelength

FAQs

What does wave-particle duality mean?

Wave-particle duality is a fundamental concept in quantum mechanics stating that every particle exhibits both wave-like and particle-like properties. Light behaves as a wave (interference, diffraction) and as particles (photons in the photoelectric effect). Electrons behave as particles (discrete charges) and as waves (electron diffraction in crystals).

Can macroscopic objects have a de Broglie wavelength?

Yes! Every object has a de Broglie wavelength, but for large objects it is unimaginably small. For a 1 kg ball moving at 1 m/s: λ = 6.626 × 10⁻³⁴ m — about 10¹⁹ times smaller than a proton. This is why quantum effects are invisible in everyday life.

Why is the de Broglie wavelength important?

The de Broglie wavelength is crucial for understanding:

• Electron microscopy — electrons have shorter wavelengths than visible light, enabling atomic-scale imaging
• Electron diffraction — used to determine crystal structures
• Quantum confinement — when a particle is confined to a space comparable to its wavelength, quantum effects dominate
• Atomic orbitals — electrons in atoms form standing waves, explaining the discrete energy levels

What is the Planck constant?

The Planck constant h = 6.62607015 × 10⁻³⁴ J·s is one of the fundamental constants of nature. It sets the scale of quantum effects. The reduced Planck constant ℏ = h / (2π) = 1.054571817 × 10⁻³⁴ J·s is also widely used.