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Walk-through

How to Use This Calculator

3 steps

1

Pick your known inputs

Use the From Mass & Velocity tab if you know a particle's mass and speed, From Momentum if momentum is already known, or From KE if you know the particle's kinetic energy (in electronvolts) — as with an electron accelerated through a voltage.

2

Enter the values

Type mass in kilograms, velocity in meters per second, momentum in kg·m/s, or kinetic energy in electronvolts — scientific notation (e.g. 9.10938e-31) is accepted directly. The calculator updates instantly and shows the de Broglie wavelength λ = h/p, auto-scaled to μm, nm, pm, or fm.

3

Read the scale interpretation

The result card explains whether the wavelength is large enough to matter — measurable and comparable to atomic dimensions for electrons and other subatomic particles, or utterly negligible for everyday macroscopic objects like a baseball.

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Reference

Formula & Methodology

3 formulas

de Broglie wavelength from momentum

λ = h / p

λ is the de Broglie wavelength in meters, h is Planck's constant (6.62607015 × 10⁻³⁴ J·s), and p is the particle's momentum in kg·m/s. This is the core relation Louis de Broglie proposed in 1924: every moving particle has an associated matter wave whose wavelength is inversely proportional to its momentum.

From mass and velocity

λ = h / (m·v)

For a non-relativistic particle, momentum p = m·v, so the wavelength can be computed directly from mass (kg) and velocity (m/s). This is the most common form taught in introductory physics.

From kinetic energy

p = √(2m·KE), so λ = h / √(2m·KE)

When kinetic energy is known instead of velocity — for example, an electron accelerated through a known voltage, where KE (eV) = accelerating voltage (V) — momentum is derived from KE = p²/(2m) first, then λ = h/p. This calculator uses the non-relativistic approximation, accurate as long as the kinetic energy stays well below the particle's rest-mass energy.

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Glossary

Key Terms Explained

6 terms

de Broglie wavelength The wavelength of the matter wave associated with a moving particle, λ = h/p. Proposed by Louis de Broglie in 1924, it extends wave-particle duality — previously used only for light — to all matter, including electrons, atoms, and (in principle) baseballs.

Matter wave The wave-like behavior exhibited by particles with mass, first confirmed experimentally by electron diffraction (Davisson–Germer, 1927). Matter waves explain phenomena like electron microscopy and quantized atomic orbitals.

Momentum (p) The product of an object's mass and velocity (p = mv) for non-relativistic speeds, measured in kilogram-meters per second (kg·m/s). Momentum, not mass or velocity alone, is what directly sets the de Broglie wavelength.

Planck's constant (h) A fundamental physical constant, exactly 6.62607015 × 10⁻³⁴ joule-seconds (fixed by the 2019 SI redefinition), that sets the scale at which quantum effects — including matter waves — become significant.

Wave-particle duality The principle that every quantum object — light or matter — exhibits both wave-like and particle-like behavior depending on how it is measured. The de Broglie wavelength quantifies the wave aspect of any massive particle.

Quantum The smallest discrete unit in which a physical quantity can change. Matter waves are a quantum-mechanical phenomenon: they become observable only when a particle's de Broglie wavelength is comparable to the size of the system it interacts with, such as a crystal lattice or an atom.

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Scenarios

Real-World Examples

3 worked examples

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Physics student

Electron in a hydrogen atom's ground-state orbit

Mass 9.10938 × 10⁻³¹ kg Velocity 2.19 × 10⁶ m/s

An electron moving at the Bohr ground-state speed has a de Broglie wavelength of about 332 pm (0.332 nm) — notably, this equals 2π times the Bohr radius, exactly matching Bohr's original quantization condition that the orbit circumference holds a whole number of wavelengths. At this scale, wave behavior is directly observable (electron diffraction, electron microscopy).

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Baseball pitcher

A 90 mph fastball

Mass 0.145 kg Velocity 40 m/s

A regulation baseball thrown at 40 m/s has a de Broglie wavelength of roughly 1.14 × 10⁻³⁴ m — vastly smaller than an atomic nucleus (~10⁻¹⁵ m), let alone the ball itself. This is why macroscopic objects never show observable wave behavior: their momentum is enormous compared to any particle physics deals with, so λ collapses to an immeasurably small number.

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Electron microscopist

An electron accelerated through 100 V (From KE tab)

Mass 9.10938 × 10⁻³¹ kg Kinetic energy 100 eV

Switching to the From KE tab and entering 100 eV (the kinetic energy gained by an electron accelerated through a 100-volt potential) gives a de Broglie wavelength of about 122.6 pm — shorter than visible light and comparable to atomic spacing in a crystal lattice, which is exactly why electron microscopes can resolve details far smaller than optical microscopes ever could.

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Deep Dive

Matter waves: understanding the de Broglie wavelength

3 chapters · 4 min read

In 1924, Louis de Broglie proposed a radical idea: if light — long understood as a wave — could also behave as discrete particles (photons), then matter, long understood as particles, should also have a wave nature. This calculator applies his relation, λ = h/p, to find the wavelength of the matter wave associated with any moving object, from a subatomic electron to a thrown baseball.

How the de Broglie Wavelength Calculator works

The calculator implements λ = h/p, where h is Planck's constant and p is the particle's momentum. If you know mass and velocity, momentum is simply p = mv (the From Mass & Velocity tab). If momentum is already known from another calculation, the From Momentum tab skips straight to λ = h/p. If instead you know kinetic energy — as when an electron has been accelerated through a known voltage — the From KE tab first recovers momentum from p = √(2m·KE), then applies the same formula.

All three tabs use the CODATA-exact value of Planck's constant, h = 6.62607015 × 10⁻³⁴ J·s, fixed by the 2019 redefinition of the SI base units.

Inputs and what they mean

Mass is entered in kilograms — for subatomic particles this means very small numbers in scientific notation (an electron's mass is 9.10938 × 10⁻³¹ kg), while everyday objects use ordinary decimal values. Velocity is in meters per second. Momentum, when entered directly, is in kilogram-meters per second (kg·m/s). Kinetic energy on the From KE tab is entered in electronvolts (eV) rather than joules, since eV is the standard unit for particle-scale energies — 1 eV is the energy an electron gains crossing a 1-volt potential difference, so an electron accelerated through 100 V has exactly 100 eV of kinetic energy.

Limits and edge cases

The From KE tab uses the non-relativistic approximation p = √(2m·KE), which is accurate as long as kinetic energy stays well below the particle's rest-mass energy (511 keV for an electron). At relativistic speeds this simplified formula underestimates momentum and overestimates wavelength — a limitation shared with most introductory-level de Broglie calculations. The formula also assumes a free, non-interacting particle; wavelengths for bound or confined particles (like electrons in an atom) are a useful conceptual guide but require full quantum mechanics for exact results. Finally, for macroscopic objects the de Broglie wavelength is so many orders of magnitude smaller than anything physically measurable that the result is a mathematical curiosity rather than an observable effect.

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Questions

Frequently Asked Questions

6 questions

What is the de Broglie wavelength formula?+

λ = h/p, where λ is the de Broglie wavelength in meters, h is Planck's constant (6.62607015 × 10⁻³⁴ J·s), and p is the particle's momentum in kg·m/s. For a non-relativistic particle, p = mv, so the formula is often written λ = h/(mv).

Why don't everyday objects show wave behavior?+

Every moving object technically has a de Broglie wavelength, but for macroscopic objects it is so small — a thrown baseball's wavelength is around 10⁻³⁴ m — that no experiment could ever detect it. Wave effects only become observable when the wavelength is comparable to the size of the system a particle interacts with, such as the spacing between atoms in a crystal, which only happens for very light, slow-moving particles like electrons.

What units does the de Broglie Wavelength Calculator expect?+

Mass is in kilograms, velocity in meters per second, momentum in kilogram-meters per second (kg·m/s), and kinetic energy (on the From KE tab) in electronvolts (eV). The result — the de Broglie wavelength — is displayed in whichever unit (μm, nm, pm, or fm) best fits the computed magnitude.

How is this different from a photon's wavelength?+

A photon's wavelength comes from its energy and frequency (E = hf = hc/λ) and photons always travel at the speed of light. The de Broglie wavelength applies to particles with mass — electrons, atoms, baseballs — moving at any speed, and uses momentum (p = mv) rather than frequency. Both are expressions of the same underlying wave-particle duality, but the formulas and physical particles involved are different.

Can I share my results?+

Yes — the calculator encodes your current tab and input values into the URL hash. Copying the address bar URL after entering your numbers is enough to share that exact scenario with a colleague or reproduce it later.

Is the From KE tab accurate for high-energy particles?+

The From KE tab uses the non-relativistic formula p = √(2m·KE), which holds well for kinetic energies far below a particle's rest-mass energy (511 keV for an electron). At relativistic energies this approximation increasingly underestimates momentum, so the calculated wavelength becomes less accurate — for those cases a full relativistic momentum-energy relation is needed instead.

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de Broglie Wavelength Calculator

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