The de Broglie wavelength calculator above computes λ = h ÷ p, the relation Louis de Broglie proposed in 1924 and was awarded the Nobel Prize in Physics for in 1929. It is a short formula with a large claim behind it: that every particle with momentum has a wavelength, not as an analogy but as a measurable physical property. Electron diffraction confirmed it experimentally within three years, and electron microscopy exists because of it.
Arb Digital publishes free physics calculators that make the boundary between similar tools explicit, because a formula with three symbols is easy to apply to the wrong quantity. This one takes the momentum of a massive particle and returns a wavelength. That is a different job from the momentum calculator, which returns momentum itself and stops there, and a different job again from the photon energy calculator, which works with light. The section below sets out exactly where each one applies.
What This De Broglie Wavelength Calculator Does
You choose a particle or type a mass, then supply whichever quantity you actually know: speed, kinetic energy in electronvolts, an accelerating voltage, or momentum directly. The tool derives the other three and reports the wavelength. Accepting four different input routes matters because real problems arrive in different forms. A diffraction experiment quotes an accelerating voltage. A scattering calculation quotes an energy. A mechanics problem quotes a speed. All three lead to the same wavelength, but only through the momentum.
The momentum model dropdown decides whether momentum is computed classically as mass times velocity or relativistically as γmv. Both are offered because both get used, and because seeing where they diverge is instructive. At one per cent of the speed of light the difference is about fifty parts per million. At half the speed of light the classical figure is wrong by fifteen per cent, and at the two hundred kilovolts used in a transmission electron microscope it is wrong by enough to move the wavelength by roughly a fifth.
The grid reports momentum, kinetic energy, speed as a fraction of the speed of light, and the wavelength again in ångström. That last unit exists on this page because ångström is still the working unit of crystallography and atomic spacing, and comparing a wavelength directly against an atomic spacing is the whole point of the calculation.
How to Use It
- Select the particle first. It fills the rest mass from the CODATA values. If you are working with a molecule, a nanoparticle or something macroscopic, choose custom and type the mass in kilograms.
- Choose the quantity you actually have. Do not convert your known value into a speed by hand before entering it. Pick the matching mode and let the tool do the conversion, because the relativistic conversions are easy to get wrong.
- Use accelerating voltage for electron optics. An electron gun at 100 V gives a wavelength near 0.123 nm. This mode assumes a single elementary charge, so double the voltage you enter for an alpha particle to get the right energy.
- Leave the model on relativistic unless you are checking a textbook answer. The relativistic form is correct everywhere and reduces to the classical one automatically at low speed, so there is no accuracy cost to using it.
- Compare the wavelength against a length that matters. Atomic spacing in a crystal is a few tenths of a nanometre. A wavelength far larger than that will not diffract usefully; a wavelength far smaller gives very small diffraction angles.
The Formula: How the De Broglie Wavelength Is Calculated
The relation is λ = h ÷ p, where h is the Planck constant and p is the momentum of the particle. It appears in exactly that form in the OpenStax University Physics Volume 3 chapter 6 key equations, alongside the matter-wave frequency relation. The Planck constant has been an exactly defined quantity since the 2019 revision of the SI, fixed at 6.626 070 15 × 10−34 joule per hertz in the NIST CODATA listing, so it contributes no uncertainty to any result on this page.
Everything therefore rests on the momentum. Classically p = mv. Relativistically p = γmv with γ = 1 ÷ √(1 − v²÷c²). When kinetic energy is the known quantity, the exact route is through the total energy: E = K + mc², then p = √(E² − (mc²)²) ÷ c. The classical shortcut p = √(2mK) is what the classical mode uses, and it is a good approximation only while the kinetic energy is small compared with the rest energy.
Work the default through. An electron has a rest mass of 9.109 × 10−31 kg. At 1.0 × 106 m/s, which is about a third of one per cent of the speed of light, the classical momentum is 9.109 × 10−25 kg·m/s. Divide the Planck constant by that and the wavelength is 7.27 × 10−10 m, or 0.727 nm, or 7.27 ångström. The relativistic momentum at that speed is larger by five parts per million, which moves the wavelength in the sixth significant figure and nowhere earlier.
Now try the accelerating-voltage mode at 100 V. The kinetic energy is 100 eV, which is 1.602 × 10−17 J. The classical momentum √(2mK) comes to 5.40 × 10−24 kg·m/s, giving a wavelength of 0.1227 nm. That is the familiar result behind the shortcut that an electron's wavelength in nanometres is roughly 1.226 divided by the square root of the accelerating voltage.
How This Differs From the Momentum and Wavelength Calculators
Three tools on this site touch these symbols and they do not overlap. The momentum calculator computes p = mv and answers questions about collisions, impulse and conservation. It ends where classical mechanics ends. Its output can be fed into this page, but it never produces a wavelength, because momentum and wavelength are different physical quantities linked only by the Planck constant.
The wavelength calculator relates the wavelength of a travelling wave to its frequency and propagation speed through λ = v ÷ f. That is classical wave mechanics and it applies to sound, water waves, light and radio. It has nothing to say about a particle's mass, because the wave it describes is not a matter wave.
The photon energy calculator handles E = hf for light. A photon has no rest mass, so its momentum is E ÷ c and its wavelength follows from that relation rather than from this one. Feeding a photon into this page by entering a mass of zero produces an undefined result, correctly, because λ = h÷p with p = mv collapses when the mass is zero.
The distinction in one sentence: this page turns the momentum of a particle that has mass into a wavelength. If you already have a wavelength and want a frequency, or you are working with light, one of the other two is the right tool.
Why an Everyday Object Has No Observable Wavelength
Run the everyday-object preset and look at the exponent. A 145-gram object moving at 40 m/s has a momentum of about 5.8 kg·m/s and therefore a de Broglie wavelength around 10−34 m. That is roughly twenty orders of magnitude smaller than a proton. There is no aperture, no crystal and no slit in the universe small enough to diffract it.
This is the honest answer to why quantum behaviour is not part of ordinary experience, and it is more satisfying than saying quantum mechanics only applies to small things. The formula applies universally. The wavelength of a macroscopic object is simply so far below any length scale it could interact with that no wave effect is ever observable. The transition is gradual, not a boundary: matter-wave interference has been demonstrated for molecules of many hundreds of atoms, at wavelengths of a few picometres, by cooling them and using very fine gratings.
The practical reading of the number is therefore comparative. A wavelength is only interesting relative to the size of the thing it meets. Electrons at a hundred volts have wavelengths matched to atomic spacing, which is why low-energy electron diffraction maps crystal surfaces. Electrons at two hundred kilovolts have wavelengths near two picometres, far below atomic spacing, which is why transmission electron microscopes resolve individual atomic columns.
Where the Relativistic Correction Starts to Bite
The classical and relativistic momenta differ by the Lorentz factor, so the size of the error depends only on speed. At 1 per cent of the speed of light the factor is 1.00005 and the classical answer is fine for any purpose. At 10 per cent it is 1.005, a half-per-cent error that matters in precision work. At 50 per cent it is 1.155, and the classical wavelength is wrong by more than a seventh.
Electron microscopy sits squarely in the region where it matters. A 200 kV electron has a kinetic energy of 200 keV against a rest energy of 511 keV, so the relativistic correction is large. The classical calculation gives about 2.74 pm and the correct relativistic value is about 2.51 pm — a difference of nearly nine per cent, which is far too large to ignore when the wavelength sets the resolution limit of the instrument. Switch the model dropdown between the two settings at that energy and the gap is immediately visible.
Protons and heavier particles need much higher energies before the correction appears, simply because their rest energy is so much larger. A proton needs almost a gigaelectronvolt of kinetic energy before relativistic effects match what an electron sees at half a megaelectronvolt. The kinetic energy calculator covers the classical energy side of that comparison, and the energy converter moves between electronvolts and joules if your source quotes one and you need the other.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the classical momentum at high energy — above a few per cent of the speed of light the classical figure is measurably wrong, and in electron microscopy it is wrong by several per cent.
- Confusing accelerating voltage with kinetic energy in volts — they coincide only for a singly charged particle. An alpha particle carries two elementary charges and gains twice the energy from the same voltage.
- Applying this formula to photons — light has no rest mass, so its momentum comes from its energy divided by the speed of light, not from mass times velocity.
- Reading the wavelength in isolation — a number in metres means nothing until you compare it with the size of whatever the particle interacts with.
- Mixing units of mass — the mass field expects kilograms. Atomic mass units and electronvolts of rest energy have to be converted first, or the exponent will be off by orders of magnitude.
Related Free Tools From Arb Digital
For classical momentum on its own use the momentum calculator, and for the classical energy of a moving mass the kinetic energy calculator. Wave relationships for light, sound and radio go through the wavelength calculator and the frequency converter, while light quanta are handled by the photon energy calculator. Energies in electronvolts and joules convert with the energy converter, and diffraction from an ordered structure is covered by the diffraction grating calculator. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
It is the wavelength associated with a particle that has momentum, given by the Planck constant divided by that momentum. Louis de Broglie proposed it in 1924 and received the 1929 Nobel Prize in Physics for the idea, which electron diffraction experiments confirmed shortly afterwards.
The momentum calculator returns momentum from mass and velocity and stops there. This page takes that momentum and converts it into a wavelength using the Planck constant. Momentum and wavelength are different quantities, and only this relation links them.
Because its momentum is enormous compared with the Planck constant, so the wavelength comes out around ten to the minus thirty-four metres. No physical structure is small enough to diffract something with that wavelength, so the wave behaviour is real in principle and unobservable in practice.
Once the speed passes a few per cent of the speed of light. At two hundred kilovolts, a common electron microscope setting, the classical result is wrong by nearly nine per cent, which is far more than the instrument's resolution tolerance.
No. A photon has no rest mass, so mass times velocity gives zero and the formula breaks down. A photon's momentum is its energy divided by the speed of light, and the photon energy calculator handles that case.
About 0.123 nanometres, or 1.23 ångström. That is close to the spacing between atoms in a crystal, which is exactly why low-energy electrons diffract from crystal surfaces and can be used to map them.
It assumes one elementary charge, so it is directly correct for electrons and protons. An alpha particle carries two elementary charges and gains twice the energy from the same voltage, so enter double the voltage to get the right result.
This tool is provided for educational and study use. It applies the published de Broglie relation with CODATA constants and does not model interference geometry, beam optics or instrument aberrations, so treat its output as a physics result rather than an instrument specification.
