de Broglie Wavelength Calculator

de Broglie relation

Every particle of momentum p has an associated wavelength inversely proportional to p:

\lambda = (h)/(p)

Non-relativistic form

For a particle of mass m and speed v (much less than c), momentum p = m v, so

\lambda = (h)/(m \, v)

Thermal de Broglie wavelength

For a gas of mass m at temperature T, the typical de Broglie wavelength is the "thermal wavelength":

\Lambda = (h)/(√(2 π m k_B T))

Why everyday objects don't diffract

A 1 kg ball at 1 m/s has λ = 6.6×10⁻³⁴ m, far smaller than any subatomic distance. Real macroscopic objects have such short wavelengths that wave behavior is unmeasurable — this is why classical mechanics works at everyday scales.

Electron diffraction

A beam of electrons accelerated through ~100 V has de Broglie wavelength on the order of 0.1 nm — about the size of an atom. This is why electron microscopes resolve structures a thousand times finer than visible-light microscopes.

Why Macroscopic Objects Show No Wave Behaviour

Planck's constant is about 6.626×10^-34\ \mathrmJ\,s, and it sits in the numerator. Any everyday momentum therefore produces a wavelength far too small to observe.

A 150 g cricket ball at 40 m/s has momentum 6 kg·m/s and a de Broglie wavelength around 1.1×10^-34 m — roughly twenty orders of magnitude smaller than a proton. No aperture could ever diffract it, so the ball follows a classical trajectory. An electron, by contrast, has a mass of only 9.11×10^-31 kg, so even at modest speeds its wavelength lands in the range of atomic spacings, where crystal lattices act as ready-made diffraction gratings. The transition between quantum and classical behaviour is not a change in the physics; it is simply that h is small.

\lambda = (h)/(p) = (h)/(mv)

Wavelength from Accelerating Voltage

For a charged particle accelerated from rest through a potential difference V, the kinetic energy is qV, so the momentum is p = √(2mqV) and the wavelength follows directly. For electrons this yields a convenient shortcut: \lambda ≈ √(1.5/V) nanometres, with V in volts.

So 100 V gives about 0.123 nm, 10 kV about 0.0123 nm, and the 100–300 kV of a transmission electron microscope gives a few picometres — though above roughly 50 kV a relativistic correction becomes necessary, since the electrons are moving at an appreciable fraction of c. This relationship is the design equation for electron optics: raising the accelerating voltage shortens the wavelength and improves the diffraction-limited resolution, at the cost of more radiation damage to the specimen.

\lambda = (h)/(√(2mqV))

Experimental Confirmation and Matter-Wave Limits

Davisson and Germer confirmed the hypothesis in 1927 by scattering electrons off a nickel crystal and finding diffraction peaks exactly where the de Broglie wavelength predicted; G. P. Thomson independently demonstrated it with thin foils. The irony is often noted that J. J. Thomson won a Nobel Prize for showing the electron is a particle and his son won one for showing it is a wave.

Since then, matter-wave interference has been demonstrated with neutrons, whole atoms, and molecules as large as fullerenes and beyond — in each case with wavelengths measured in picometres. Neutron diffraction is now a routine tool for locating hydrogen atoms in crystals, where X-rays struggle. The practical ceiling is not any sudden failure of the relation but decoherence: the larger the object, the more it interacts with its surroundings, and the more quickly the phase relationship that makes interference visible is destroyed.

Worked Example

Inputs

• mass: 9.1093837015e-31
• velocity: 10000000

Result: wavelength: 7.274e-11 wavelengthNm: 0.0727 momentum: 9.109e-24

Explanation

An electron (9.11×10⁻³¹ kg) accelerated to 10⁷ m/s has momentum p = 9.11×10⁻²⁴ kg·m/s and de Broglie wavelength λ = 6.626×10⁻³⁴ / 9.11×10⁻²⁴ ≈ 7.27×10⁻¹¹ m = 0.073 nm — small enough to resolve single atoms.

Second Scenario

Inputs

• mass: 1
• velocity: 10000000

Result: wavelength: 7.274e-11 wavelengthNm: 0.0727 momentum: 9.109e-24

Explanation

This scenario uses different inputs (mass = 1, velocity = 10000000) to show how changing one variable affects the de broglie wavelength result. Run the calculator above with these values to get the exact updated output with step-by-step work.