De Broglie Wavelength Calculator - Mass and Speed to λ

A de broglie wavelength calculator turns any particle's rest mass and motion into the matter-wave wavelength λ = h / p, where p is the relativistic momentum γ m v. Enter a mass and a speed, and the calculator returns the wavelength in m, pm, nm, Å, and fm with the momentum and Lorentz factor.

• • Modern physics homework: Solve textbook problems that ask for the de Broglie wavelength of a 100 eV electron, a 200 MeV proton, or a 40 m/s baseball.
• • Electron diffraction setup: Pick a TEM accelerating voltage, get the matter-wave wavelength in picometres, and decide whether the beam resolves a chosen lattice spacing.
• • Neutron and atom optics: Estimate the wavelength of cold neutrons, slow atoms, and molecule beams to size mirrors, gratings, and interferometer arms.
• • Relativistic sanity check: Watch γ grow above 1.0 as the speed climbs past 0.1 c, and use the relativistic momentum in λ = h / p.

The de Broglie wavelength is the wavelength that Louis de Broglie assigned to every moving particle in his 1924 thesis, and λ = h / p sets the spatial resolution of an electron microscope.

The de Broglie wavelength feeds straight into the wave side of the same intro-physics toolkit, and the Harmonic Wave Equation Calculator returns the displacement-versus-position profile for any sinusoidal wave.

The calculator combines λ = h / p with the relativistic momentum p = γ m v, where γ = 1 / sqrt(1 - v² / c²). The user supplies a mass plus a state input, and the formula returns the wavelength, momentum, and Lorentz factor.

• massValue / massUnit: Numerical particle mass and unit selector (kg, g, mg, μg, m_e, m_p, or u), converted to kilograms internally.
• inputMode: Switches between speed (m/s) and kinetic energy (eV).
• velocityValue / energyValue: Speed in m/s, or kinetic energy in eV. Rejected at v ≥ c or negative values.
• De Broglie wavelength λ: Output in m, pm, nm, Å, and fm, equal to h / (γ m v).
• Momentum p: Output in kg·m/s, equal to γ m v.
• Lorentz factor γ: Dimensionless. Reads 1.0 in the non-relativistic limit and grows as v approaches c.

Every output is a direct evaluation of the de Broglie relation with the same NIST CODATA constants that a textbook would quote. A speed of zero gives an undefined wavelength rather than a division-by-zero error.

According to NIST CODATA 2018 - Planck constant, Planck constant h = 6.62607015 × 10^-34 J·s

The Compton wavelength is the companion to the de Broglie wavelength for a particle at rest, and the Compton Wavelength Calculator returns the same NIST constants in the rest-mass case.

Four ideas drive every output: the de Broglie relation, relativistic momentum, the Lorentz factor at high speed, and the matter-wave diffraction link.

Together these four ideas turn the de Broglie wavelength from a one-line formula into the practical input for a diffraction experiment or particle interferometer. As published by Omni Calculator, the de Broglie wavelength is typically entered as a particle mass with either a speed in m/s or a kinetic energy in eV, matching the modern physics textbook workflow.

For the orbital-wavelength side of the same atomic-physics course unit, the Bohr Model Calculator returns the hydrogen-like transition wavelength that the de Broglie relation has to match for a stationary orbit.

Enter the particle mass, pick a state input mode, and read the seven outputs. The page recalculates as you type.

1. 1 Enter the particle mass: Type the mass as a positive number. Pick the unit that matches the value (kg for SI, m_e for atomic-scale, m_p for nuclear-scale, u for nucleons).
2. 2 Pick the state input mode: Use velocity for m/s, or kinetic energy for eV. The calculator hides the inactive field.
3. 3 Type the velocity or kinetic energy: For velocity, enter a positive number below c. For kinetic energy, enter a non-negative electronvolt value.
4. 4 Read the de Broglie wavelength: The first result is the wavelength in metres. The follow-up rows give pm, nm, Å, and fm for the same particle.
5. 5 Check the momentum and gamma: Momentum in kg·m/s and the Lorentz factor γ are the two physical inputs. γ = 1.0 in the classical limit.
6. 6 Reset and try another particle: Use Reset to return to 1 m_e at 5.93e6 m/s. Switch the mass unit from m_e to m_p to see the same form on a proton.

Leave the mass at 1 and switch the unit from m_e to m_p with 5.93e6 m/s. The de Broglie wavelength drops from about 0.12 nm to about 6.6e-14 m (0.066 pm, 0.66 Å) because the heavier proton has 1836 times the rest mass.

When the same problem asks for the recoil target's momentum rather than the wavelength, the Conservation of Momentum Calculator returns the velocity split for an elastic collision using the same p = m v input.

The de broglie wavelength calculator is most useful when a modern physics, quantum, or diffraction problem asks for the matter-wave wavelength of a particle and the student or engineer wants the numerical answer in seconds.

• • NIST CODATA constants: Uses Planck constant, speed of light, electron mass, proton mass, and atomic mass unit from NIST CODATA 2018, matching textbook values to at least nine significant figures.
• • Velocity or kinetic energy: Accepts a speed in m/s or a kinetic energy in eV. The energy mode uses the relativistic E² = (pc)² + (mc²)² relation.
• • Five wavelength units: Returns the de Broglie wavelength in m, pm, nm, Å, and fm, so the same answer works for atomic-scale electrons and nuclear-scale protons.
• • Relativistic momentum built in: Uses p = γ m v with γ reported alongside the wavelength, so it is obvious when the relativistic correction matters.
• • Edge-safe inputs: Rejects negative mass, zero momentum, and v ≥ c with a clear error. Defaults to 1 m_e at 5.93e6 m/s, reproducing the textbook 0.727 nm electron wavelength.

For atomic and subatomic problems the de Broglie wavelength is the right length to quote, and the same form covers electron beams in a TEM, slow neutrons in a guide, and proton beams in a cyclotron.

When the kinetic energy that goes into the de Broglie formula needs to be checked against the textbook value, the Work-Energy-Power Calculator converts between joules, electronvolts, and watts for the same particle.

Two input factors set every output, and three physical constraints decide when the de Broglie wavelength is the right length scale to quote.

• • The calculator assumes a single particle of well-defined momentum. For an ensemble with a momentum spread, the matter wave is a wave packet and the spectral width of the wavelength matters more than any single value.
• • At zero momentum the de Broglie wavelength diverges. The calculator reports the wavelength as undefined for v = 0 or E_K = 0.
• • At very low mass with very high speed, γ becomes very large and the wavelength can become numerically tiny. The calculator clamps to a finite gamma but the wavelength can drop below any measurable scale.

For textbook problems on electrons, protons, neutrons, atoms, and small molecules the de Broglie wavelength is exact to the precision of the NIST CODATA constants.

As published by Wikipedia - Matter wave, De Broglie wavelength uses relativistic momentum p = γ m v

As published by Omni Calculator - De Broglie wavelength, Modern physics textbook de Broglie wavelength form pattern (mass + velocity, mass + energy)

When the moving particle is a photon rather than a massive object, the Compton Scattering Calculator uses the same p = h / λ on the photon side and returns the wavelength shift of a Compton scattering event.