⚛️ De Broglie Wavelength Calculator
Calculate the de Broglie wavelength of any particle using λ = h/p. Solve for wavelength, momentum, mass, velocity, or kinetic energy. Includes relativistic corrections and common particle presets.
What Is the De Broglie Wavelength?
The de Broglie wavelength is the wavelength associated with a material particle in quantum mechanics. It was proposed by the French physicist Louis de Broglie in his 1924 doctoral thesis, in which he hypothesized that all matter — not just light — has wave-like properties.
The relationship is elegantly simple:
λ = h / p
where λ = de Broglie wavelength (m), h = Planck constant (6.62607015 × 10⁻³⁴ J·s), and p = particle momentum (kg·m/s)
For a non-relativistic particle, momentum is p = mv, so the formula becomes λ = h / (mv). If the kinetic energy K is known instead, p = √(2mK) gives λ = h / √(2mK).
Wave-Particle Duality
The de Broglie hypothesis is a cornerstone of wave-particle duality — the idea that every quantum entity exhibits both wave and particle properties. Just as light (a wave) can behave as a particle (photon), particles of matter can behave as waves.
The wavelength is inversely proportional to momentum:
- Heavier particles → shorter wavelength
- Faster particles → shorter wavelength
- Lighter, slower particles → longer wavelength (more wave-like)
For a baseball (150 g moving at 40 m/s), the de Broglie wavelength is approximately 1.1 × 10⁻³⁴ m — about 10²⁰ times smaller than a proton, making wave effects undetectable. For an electron accelerated through 100 V (≈ 100 eV), the wavelength is 0.123 nm — comparable to atomic spacings, which is why electron diffraction is observable.
Experimental Confirmation
The de Broglie hypothesis was confirmed in 1927 by the Davisson-Germer experiment, which observed electron diffraction from a nickel crystal. The measured wavelengths matched de Broglie's predictions exactly. G.P. Thomson independently confirmed electron diffraction using gold foil the same year. De Broglie received the 1929 Nobel Prize in Physics for this work.
Since then, diffraction has been observed for:
- Electrons (1927) — Davisson, Germer, Thomson
- Neutrons (1930s) — used in neutron diffraction for crystallography
- Atoms and molecules (1930s–1990s) — helium, hydrogen, and even large molecules like C₆₀
- Large molecules (2019) — up to 25,000 atomic mass units (the "matter-wave interference" experiments)
Relativistic Corrections
When a particle's velocity approaches a significant fraction of the speed of light (c ≈ 3 × 10⁸ m/s), the classical momentum p = mv must be replaced by the relativistic momentum p = γmv, where:
γ = 1 / √(1 − v²/c²)
The relativistic de Broglie wavelength is λ = h / (γmv)
Relativistic effects become important when the particle's velocity exceeds about 10% of the speed of light (v > 0.1c ≈ 3 × 10⁷ m/s). This calculator automatically detects relativistic speeds and displays a warning.
Why Use This Calculator?
- Solve for any variable: Calculate λ, p, m, or v depending on what's unknown.
- Kinetic energy option: Calculate wavelength from kinetic energy using p = √(2mK).
- Particle presets: Quickly select electron, proton, neutron, alpha particle, or muon.
- Relativistic warning: Automatically flags when velocities approach the speed of light.
- Step-by-step solution: See exactly how the calculation is performed.
- Free & private: No registration, no data storage — all calculations are done in your browser.
❓ De Broglie Wavelength Calculator FAQ
What is the de Broglie wavelength formula?
The formula is λ = h / p, where λ is the de Broglie wavelength, h is Planck's constant (6.62607015 × 10⁻³⁴ J·s), and p is the particle's momentum. For a non-relativistic particle, p = mv, so λ = h/(mv).
What is Planck's constant?
Planck's constant is a fundamental physical constant: h = 6.62607015 × 10⁻³⁴ J·s. It relates the energy of a photon to its frequency (E = hf) and appears throughout quantum mechanics.
What is the de Broglie wavelength of an electron?
It depends on the electron's speed. For an electron accelerated through 100 V (kinetic energy ≈ 100 eV), the de Broglie wavelength is approximately 0.123 nm. At 1 keV, it's about 0.0388 nm, and at 100 keV, about 0.00370 nm.
Why don't we observe wave properties in everyday objects?
Because the de Broglie wavelength is inversely proportional to momentum. Macroscopic objects have huge masses and momenta, so their wavelengths are immeasurably small — typically 10⁻³⁴ m or smaller, far below any measurable scale.
What is the difference between the classical and relativistic de Broglie wavelength?
The classical formula uses p = mv. The relativistic formula uses p = γmv, where γ = 1/√(1 − v²/c²). Relativistic corrections are needed when v > 0.1c (about 3 × 10⁷ m/s).
How does the de Broglie wavelength relate to the Bohr model?
De Broglie's hypothesis explained Bohr's quantization condition: electrons in stable orbits have an integer number of wavelengths fitting around the orbit circumference: 2πr = nλ. This was a major success of the de Broglie hypothesis.
What is the wavelength of a photon?
For a photon, which has zero rest mass, the momentum is p = E/c = hf/c = h/λ. So the de Broglie formula λ = h/p gives the same wavelength as the classical electromagnetic wave: λ = hc/E = c/f.
Can large molecules exhibit de Broglie wave behaviour?
Yes! Interference has been observed for molecules as large as 25,000 atomic mass units (about 25,000 times the mass of a hydrogen atom). These experiments push the boundary between quantum and classical physics.
How accurate is this calculator?
This calculator uses the standard de Broglie formula with precise physical constants. For non-relativistic velocities (v < 0.1c), the classical formula is highly accurate. For relativistic speeds, the calculator warns you and shows the classical approximation — for exact results, use the relativistic formula λ = h/(γmv).
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required.