de Broglie Wavelength Calculator

Wave nature of moving matter

de Broglie proposed that every moving particle has a wavelength linked to momentum. When momentum increases, the wavelength shrinks. This calculator helps compare macroscopic objects with quantum particles by showing how quickly λ becomes extremely small for everyday masses and speeds. For example, doubling speed halves wavelength, and doubling mass halves it too, showing the simple inverse dependence on p.

Core equation and constants

The calculator uses λ = h/p with h = 6.62607015×10⁻³⁴ J·s. If you provide momentum directly, it converts your unit to kg·m/s before dividing. If you provide mass and velocity, it computes p = m v. For mass with kinetic energy, it uses p = √(2mK) for the non‑relativistic case.

Scale of wavelengths in practice

Typical wavelengths span huge ranges. A 1 kg object moving at 1 m/s has λ ≈ 6.6×10⁻³⁴ m, far beyond direct detection. A thermal neutron (m ≈ 1.675×10⁻²⁷ kg) at 2200 m/s gives λ around 1.8 Å, comparable to crystal spacings, which is why neutrons diffract in solids.

Voltage accelerated electrons

For electrons accelerated through a voltage, kinetic energy is K = eV. At 150 V the wavelength is on the order of 1 Å, useful for electron diffraction demonstrations. At 1 kV it drops further, and at 10 kV it reaches tenths of an angstrom, enabling very high spatial resolution in electron optics.

When relativistic momentum matters

Relativistic effects matter when kinetic energy approaches the rest‑energy scale. For electrons, mc² is 511 keV, so corrections become noticeable from a few kilovolts upward. The calculator’s relativistic options use p = √(K² + 2Kmc²)/c or p = γmv, ensuring λ is not underestimated at high voltage or high speed.

Interpreting the outputs

Along with λ in meters, the output includes nanometers, picometers, and angstroms. It also reports the wavenumber k = 2π/λ in rad/m, which is convenient for wave descriptions and interference calculations. If kinetic energy is known, the tool estimates a matter‑wave frequency using f = K/h. Choose Å for atomic spacing, pm for X‑ray ranges, and nm for cold atoms; the converted value helps compare scales.

Input tips and common checks

To improve reliability, keep units consistent and choose the input method that matches your measurements. Use scientific notation for very small masses or momenta. For charged particles in accelerators, prefer the voltage mode. For near‑light speeds, use the relativistic velocity mode and ensure v remains below c.

FAQs

What is the de Broglie wavelength?

It is the wavelength associated with a moving particle, defined by λ = h/p. Larger momentum means a shorter wavelength, which makes wave effects harder to observe for massive, fast objects.

Which method should I choose?

Use momentum if you already have p. Use mass and velocity for classical motion. Use mass and kinetic energy when energy is measured. Use electron voltage for accelerated electrons. Use relativistic modes for high speeds or high energies.

When should I enable relativistic correction for electrons?

Relativistic correction becomes noticeable above a few kilovolts and grows with voltage. It is important for precision at tens of kilovolts and beyond, because the non‑relativistic formula underestimates momentum and overestimates λ.

Why does the calculator show Å, nm, and pm?

These units match common physical scales. Å is convenient for atomic spacings and electron diffraction. pm suits very short wavelengths, and nm suits longer waves such as cold atoms or slow molecules.

Can I enter mass in atomic mass units?

Yes. Select u (amu) and enter the value. The calculator converts to kilograms internally using 1 u = 1.66053906660×10⁻²⁷ kg before computing momentum and wavelength.

What does the wavenumber output mean?

The wavenumber k = 2π/λ describes how rapidly the phase changes with distance. It is useful in interference, diffraction, and wave‑equation forms where k appears directly, especially when comparing different λ values.