de Broglie Wavelength Calculator
Solve the de Broglie relation λ = h / p for any of its three unknowns — wavelength, momentum, or Planck's constant — with step-by-step math, a live particle–wave schematic, and companion relativistic metrics for electrons, protons, and photons.
Every particle is a wave
In 1924 Louis de Broglie proposed that the wave–particle duality Einstein had applied to light should apply to matter as well. If a photon of momentum p has wavelength λ = h/p, then so should an electron, a proton, or a bowling ball. The relation was confirmed three years later when Davisson and Germer diffracted electrons from a nickel crystal and measured exactly the pattern the formula predicts. The wavelength is tiny for everyday objects — a 1 kg ball moving at 1 m/s has λ ≈ 6.6 × 10⁻³⁴ m, far below any measurable scale — which is why Newtonian mechanics works so well.
Why electron microscopes beat light microscopes
Optical microscopes are limited by the wavelength of visible light, roughly 500 nm. Electron microscopes use electrons accelerated to 100 keV, whose de Broglie wavelength is about 3.7 pm — over 100,000 times shorter. This is why electron microscopes can resolve individual atoms while light microscopes cannot resolve anything smaller than a bacterium. The formula λ = h/p directly tells you the resolution limit: want to see smaller things, use particles with larger momentum.
It explains Bohr's quantised orbits
De Broglie's relation gave the first physical explanation of Bohr's mysterious rule that angular momentum is quantised in units of ħ. If an electron orbits a nucleus as a standing wave, the circumference must contain a whole number of wavelengths: 2πr = nλ. Substituting λ = h/p and rearranging gives mvr = nħ — exactly Bohr's quantisation condition. The electron orbits only exist at radii where the wave closes on itself smoothly, which is why atomic energy levels are discrete.
Thermal wavelength sets the quantum regime
Every particle at temperature T has a thermal de Broglie wavelength λ_th = h/√(2πmkT). When λ_th is much smaller than the interparticle spacing, the gas behaves classically. When λ_th becomes comparable to the spacing — at very low temperatures or very high densities — the wavefunctions overlap and quantum statistics take over. This is the boundary between classical and quantum gases, and it is why helium remains liquid down to absolute zero while nitrogen freezes solid: helium atoms are light enough that their thermal wavelength is large even at 4 K.
How to Use This Calculator
The de Broglie relation links three quantities: λ, h, and p. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick λ, p, or h from the toggle. The calculator rearranges λ = h/p automatically for whichever variable you choose.
2. Use realistic presets
Wavelength presets span 1 pm to 1 μm; momentum presets span 1 eV/c to 1.9 GeV/c; the Planck constant preset is the exact CODATA value 6.62607015 × 10⁻³⁴ J·s.
3. Match units carefully
Wavelength is in metres, momentum in kg·m/s, and h in J·s — the SI combination that makes λ = h/p work directly. If you have momentum in eV/c, convert first, or use the converter card further down the page. The hint under each field shows the value in alternative units.
4. Read the result and verdict
The result card shows the solved quantity plus equivalent units. A colour-coded verdict classifies the wavelength scale from macroscopic through atomic, nuclear, and ultra-high-energy regimes.
5. Check the schematic and charts
The schematic shows a particle with momentum p on the left, the Planck bridge in the middle, and the resulting wave on the right. The basic chart plots λ against p on a log–log scale for three values of h; the advanced chart plots λ against kinetic energy for an electron, a proton, and a C₆₀ molecule.
6. Switch to Advanced for depth
Advanced mode adds the λ vs. kinetic-energy chart, a sensitivity check on a +1% input change, and companion metrics including the equivalent photon energy, the relativistic velocity and kinetic energy for an electron and a proton at the given momentum, and the thermal de Broglie wavelength at 300 K.
What This Tool Offers
Built around the de Broglie relation λ = h/p, the bridge between the particle and wave descriptions of matter.
Three-way solving
Solve for wavelength, momentum, or Planck's constant. The calculator rearranges the formula for whichever variable you choose.
Particle–wave schematic
A live diagram with the particle and its momentum arrow on the left, the Planck bridge in the middle, and a drawn sine wave with a λ period marker on the right.
Wavelength-class verdict
Each result is classified from macroscopic through microwave, infrared, visible, ultraviolet, X-ray, gamma-ray, and ultra-high-energy regimes, with a coloured badge.
Multi-unit hints
Each field shows the value converted to related units — pm, Å, and nm for wavelength; eV/c, keV/c, and MeV/c for momentum; eV·s for Planck's constant.
Dual comparison charts
A log–log λ vs. p chart for three values of h, plus an advanced λ vs. kinetic-energy chart for electron, proton, and C₆₀ that shows both the non-relativistic −½ slope and the relativistic −1 limit.
Reference tables
A momentum ↔ wavelength converter, a lookup table of λ against p with equivalent electron and proton kinetic energies, typical de Broglie wavelengths by particle, and a full variable reference table.
Reference Tables & Conversions
A momentum converter, a λ lookup grid, and typical de Broglie wavelengths by particle.
Momentum ↔ wavelength ↔ equivalent photon converter
de Broglie Wavelength Lookup
Computed from λ = h/p with h = 6.62607015 × 10⁻³⁴ J·s. Kinetic energies are relativistic.
| λ (m) | p (kg·m/s) | p (eV/c) | e⁻ kinetic energy | p⁺ kinetic energy | Scale |
|---|
Typical de Broglie Wavelengths by Particle
Representative values computed from λ = h/p using relativistic kinematics where appropriate.
| Particle / scenario | Momentum (kg·m/s) | λ (m) | Notes |
|---|---|---|---|
| Thermal neutron at 300 K | 4.93 × 10⁻²⁴ | 1.34 × 10⁻¹⁰ | ≈ 1.34 Å — matches atomic spacings, hence neutron diffraction |
| Electron at 10⁵ m/s (0.34 eV) | 9.11 × 10⁻²⁶ | 7.27 × 10⁻⁹ | Low-energy electron diffraction (LEED) range |
| Electron at 10⁶ m/s (2.84 eV) | 9.11 × 10⁻²⁵ | 7.27 × 10⁻¹⁰ | The calculator default — atomic scale |
| Electron at 100 eV (accelerated) | 5.40 × 10⁻²⁴ | 1.23 × 10⁻¹⁰ | Typical electron diffraction energy |
| Electron at 10 keV | 5.40 × 10⁻²³ | 1.23 × 10⁻¹¹ | 12.3 pm — TEM resolution range |
| Electron at 100 keV | 1.66 × 10⁻²² | 3.70 × 10⁻¹² | 3.70 pm — high-resolution TEM |
| Electron at 1 GeV | 5.34 × 10⁻¹⁹ | 1.24 × 10⁻¹⁵ | 1.24 fm — nuclear scale, deep inelastic scattering |
| Proton at 1 eV (thermal) | 2.31 × 10⁻²³ | 2.86 × 10⁻¹¹ | 28.6 pm — much shorter than an electron at the same energy |
| Proton at 1 MeV | 2.31 × 10⁻²⁰ | 2.86 × 10⁻¹⁴ | 28.6 fm — nuclear probe |
| C₆₀ buckyball at 1 eV | 6.19 × 10⁻²² | 1.07 × 10⁻¹² | ≈ 1 pm — demonstrated in 2019 matter-wave interference |
| Human at 1 m/s (70 kg) | 70 | 9.47 × 10⁻³⁶ | 36 orders of magnitude below the Planck length |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| λ | de Broglie wavelength | 10⁻¹⁵ – 10⁻³ | m |
| p | Linear momentum (γmv) | 10⁻³⁶ – 10⁻¹⁰ | kg·m/s |
| h | Planck constant | 6.62607015 × 10⁻³⁴ (exact) | J·s |
| ħ | Reduced Planck constant h/2π | 1.054571817 × 10⁻³⁴ (exact) | J·s |
| k | Wavenumber 2π/λ | 10⁰ – 10¹⁶ | rad/m |
| eV/c | Momentum unit in particle physics | — | 5.344286 × 10⁻²⁸ kg·m/s |
| mₑ | Electron mass | 9.1093837015 × 10⁻³¹ | kg |
| mₚ | Proton mass | 1.67262192369 × 10⁻²⁷ | kg |
What Does λ = h / p Measure?
The de Broglie relation connects three quantities at the foundation of quantum mechanics: the wavelength of a matter wave (λ), the momentum of the particle it describes (p), and the Planck constant (h) that sets the scale of quantum effects. Together they express the central insight of wave–particle duality — every particle with momentum p has an associated wavelength λ, and every wave with wavelength λ carries momentum p.
de Broglie Wavelength Formula
The defining relationship, proposed by Louis de Broglie in 1924, is:
Rearranged to solve for the other two unknowns:
h = λ · p
For a non-relativistic particle of mass m and velocity v, the momentum is p = mv and the wavelength becomes:
Expressed in terms of kinetic energy K = p²/(2m), the relation takes the form used throughout electron microscopy:
For a relativistic particle the momentum is p = γmv and the relation becomes λ = h / (γmv), with γ = 1/√(1 − v²/c²).
de Broglie Variables
| Symbol | Meaning | Notes |
|---|---|---|
| λ | de Broglie wavelength | Wavelength of the matter wave, in metres |
| h | Planck constant | 6.62607015 × 10⁻³⁴ J·s, exact since the 2019 SI redefinition |
| p | Momentum | p = mv non-relativistically; p = γmv relativistically |
| m | Particle mass | Rest mass, in kilograms |
| v | Velocity | Must be less than c for massive particles |
| γ | Lorentz factor | γ = 1/√(1 − v²/c²); approaches 1 for v ≪ c |
| K | Kinetic energy | K = p²/(2m) non-relativistically; K = (γ − 1)mc² relativistically |
How Each Variable Affects the Wavelength
The de Broglie wavelength is inversely proportional to momentum, so heavier and faster particles have shorter wavelengths. But because momentum itself scales linearly with mass and velocity, the wavelength scales as 1/m and 1/v. A proton, being about 1836 times more massive than an electron, has a wavelength 1836 times shorter at the same speed — and about 43 times shorter at the same kinetic energy, because the energy–momentum relation differs by the square root of the mass ratio.
This is why electron microscopes use electrons rather than protons: at the same accelerating voltage, electrons give a longer wavelength but are far easier to focus and produce in high-current beams. Neutrons, being neutral, penetrate deeply into materials and have wavelengths comparable to atomic spacings, which is why neutron diffraction is a workhorse of condensed-matter physics. And at very high energies, all particles approach the same wavelength for the same momentum, because the mass term becomes negligible and λ ≈ hc/E.
de Broglie Wavelength Calculation Examples
p = m v = 9.109 × 10⁻³¹ × 10⁶ = 9.109 × 10⁻²⁵ kg·m/s
λ = h/p = 6.626 × 10⁻³⁴ / 9.109 × 10⁻²⁵ = 7.274 × 10⁻¹⁰ m = 727.4 pm
p = h/λ = 6.626 × 10⁻³⁴ / 1.00 × 10⁻¹⁰ = 6.626 × 10⁻²⁴ kg·m/s
In eV/c: p = 6.626 × 10⁻²⁴ / 5.344 × 10⁻²⁸ = 12.4 keV/c
h = λ · p = 3.70 × 10⁻¹² × 1.79 × 10⁻²² = 6.626 × 10⁻³⁴ J·s
E = (3/2) kT ≈ 0.0388 eV → p = √(2mK) = 4.93 × 10⁻²⁴ kg·m/s
λ = h/p = 1.34 × 10⁻¹⁰ m ≈ 1.34 Å
Frequently Asked Questions
Wave–particle duality is the quantum-mechanical statement that every physical entity — light, electrons, atoms, even large molecules — exhibits both wave and particle properties, depending on the experiment. A beam of electrons can diffract like a wave and interfere with itself, but each electron arrives at the detector as a single localised impact. De Broglie's relation λ = h/p is the quantitative bridge between the two descriptions: it tells you what wavelength to assign to a particle of given momentum. The wave is not a physical vibration in a medium — it is a probability amplitude whose squared modulus gives the probability of finding the particle at a given place.
Because the Planck constant is extraordinarily small. A 1 kg ball moving at 1 m/s has momentum 1 kg·m/s, so its de Broglie wavelength is 6.626 × 10⁻³⁴ m — thirty-three orders of magnitude smaller than an atomic nucleus. No conceivable experiment could detect such a wavelength, so the ball behaves purely classically. Quantum effects become visible only when the wavelength is comparable to the size of the system: electrons in atoms (λ ~ 10⁻¹⁰ m), neutrons in crystals (λ ~ 10⁻¹⁰ m), or ultracold atoms in traps (λ ~ 10⁻⁶ m). This is the correspondence principle in action.
For a non-relativistic particle use K = p²/(2m), so p = √(2mK) and λ = h/√(2mK). This is the formula used throughout electron microscopy: for an electron accelerated through a potential V, K = eV, so λ = h/√(2mₑeV) ≈ 1.226 nm / √(V in volts). For relativistic particles use K = (γ − 1)mc² and solve for p: pc = √(K² + 2Kmc²). The relativistic formula reduces to the non-relativistic one when K ≪ mc² — below about 10 keV for electrons, or 1 GeV for protons.
It is a real wave in the sense that it produces measurable interference and diffraction patterns, but it is not a wave of material displacement like a sound wave or water wave. In the standard interpretation, the de Broglie wave is a probability amplitude: the complex-valued wavefunction ψ(x,t) whose squared modulus |ψ|² gives the probability density of finding the particle at position x. Where the amplitude is large, the particle is likely to be detected; where it is small, detection is unlikely. The wave itself is not directly observable — only its intensity pattern, accumulated over many particles, is.
The de Broglie wavelength λ = h/p depends on how fast the particle is moving; the Compton wavelength λ_C = h/(mc) depends only on the particle's rest mass. For an electron, λ_C = 2.426 pm — a fixed quantum-mechanical length scale that appears in Compton scattering and in the Klein–Nishina cross-section. The de Broglie wavelength equals the Compton wavelength when the particle is moving at the speed of light, i.e. p = mc. For an electron that occurs at a kinetic energy of about 212 keV. In particle physics the reduced Compton wavelength ħ/(mc) sets the range of the corresponding force, which is why the weak force is short-ranged (heavy W boson) while electromagnetism is long-ranged (massless photon).
Directly. The de Broglie relation p = h/λ = ħk, combined with the Fourier relationship between position and wavenumber, gives the Heisenberg uncertainty principle Δx · Δp ≥ ħ/2. A wave packet that is sharply localised in space necessarily contains a wide range of wavenumbers, hence a wide range of momenta. Conversely, a particle with a well-defined momentum has a wavelength that extends over all space. The de Broglie relation is the kinematic foundation of the uncertainty principle, and it explains why quantum mechanics is inherently probabilistic — you cannot simultaneously localise a particle and give it a precise momentum.
The thermal de Broglie wavelength λ_th = h/√(2πmkT) is the typical de Broglie wavelength of a particle in a gas at temperature T. It sets the boundary between classical and quantum behaviour: when λ_th is much smaller than the average interparticle spacing, the gas is classical and Maxwell–Boltzmann statistics apply. When λ_th becomes comparable to the spacing — at low temperature or high density — the wavefunctions overlap and Bose–Einstein or Fermi–Dirac statistics take over. For helium-4 at 4 K, λ_th ≈ 0.4 nm, comparable to the interatomic spacing, which is why liquid helium exhibits superfluidity. For air at room temperature, λ_th ≈ 0.02 nm, far smaller than the molecular spacing, so air is classical.
Divide the momentum in kg·m/s by the elementary charge divided by the speed of light: p[eV/c] = p[kg·m/s] / (e/c), where e/c = 1.602176634 × 10⁻¹⁹ / 299792458 = 5.344286 × 10⁻²⁸. So 1 eV/c = 5.344286 × 10⁻²⁸ kg·m/s. Going the other way, multiply by 5.344286 × 10⁻²⁸. This conversion is used throughout particle physics because the natural units set c = 1, making momentum and energy the same dimension. The converter card above performs this conversion automatically.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| lambda | de Broglie wavelength | m |
| h | Planck's constant | 6.626e-34 J.s |
| p | Momentum of the particle | kg.m/s |
What it means
The de Broglie wavelength is the wavelength associated with a moving particle, demonstrating that matter has wave‑like properties. The formula λ = h/p relates the wavelength to momentum. This concept is central to quantum mechanics and explains phenomena like electron diffraction and the quantization of atomic orbits. It is used in electron microscopy, where electrons are used to image objects at atomic scale. The de Broglie wavelength also appears in the Heisenberg uncertainty principle and in the development of wave mechanics. Understanding this duality is fundamental to modern physics.
Worked example
de Broglie Wavelength – Two Examples
Real‑World| Parameter | Value |
|---|---|
| p | 1×10⁻²⁴ kg·m/s |
| Parameter | Value |
|---|---|
| p | 5×10⁻²⁴ |
Common mistakes
- Momentum p: For a particle – p = mv (non‑relativistic) or relativistic.
- de Broglie wavelength λ: In metres – typically very small for macroscopic objects.
- Planck’s constant h: 6.626×10⁻³⁴ J·s.
- Wave‑particle duality: Applicable to all matter, not just subatomic particles.
- Units: p in kg·m/s → λ in m.
Applications
The de Broglie wavelength, λ = h/p, associates a wavelength with a particle, highlighting the wave‑particle duality. This concept is essential in quantum mechanics and is used in electron microscopy, neutron diffraction, and particle physics. Engineers apply it in the design of electron beam lithography, in scanning tunnelling microscopes, and in materials characterisation. The wavelength of electrons determines the resolution in electron microscopes. By understanding de Broglie's relation, professionals can exploit wave properties of matter to achieve high‑resolution imaging and to study the structure of matter at the atomic level.
- Design of electron microscopes (TEM, SEM)
- Electron beam lithography for semiconductor fabrication
- Neutron diffraction for crystal structure analysis
- Scanning tunnelling microscopy (STM)
- Quantum interference and atom optics
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