The de Broglie wavelength calculator lets you compute the quantum mechanical wavelength of any moving particle — electron, proton, neutron, alpha particle, or a custom mass — in a single click. It implements the real de Broglie relation with the exact CODATA value of Planck’s constant (h = 6.62607015 × 10⁻³⁴ J·s) and can rearrange the formula in four ways: solve for wavelength, momentum, velocity, or mass. Every result comes with a full substitution line showing the numbers and units at each step.
The physics behind the formula
In classical physics, waves are waves and particles are particles. In 1924 Louis de Broglie shattered that clean separation. He reasoned by symmetry: if light — a wave — can behave like a particle (the photon), then particles of matter should behave like waves. The wavelength he assigned to a particle of momentum p = mv is:
λ = h / p = h / (m · v)
where h = 6.626 × 10⁻³⁴ J·s is Planck’s constant, m is the particle’s rest mass in kilograms, and v is its speed in metres per second.
The same equation rearranges immediately into three other useful forms:
- Momentum from wavelength: p = h / λ
- Velocity from mass and wavelength: v = h / (m · λ)
- Mass from wavelength and velocity: m = h / (λ · v)
The calculator covers all four rearrangements and converts between metres, nanometres, picometres, Ångströms, and femtometres automatically.
How it works
- Select what you want to find from the Solve for dropdown.
- If the solved quantity is wavelength, velocity, or mass, choose a particle preset (electron, proton, neutron, or alpha particle) — the rest mass is filled in from CODATA values. Or choose Custom mass and type your own value in kg or atomic mass units (u, where 1 u = 1.66054 × 10⁻²⁷ kg).
- Enter the velocity (m/s, km/s, or as a fraction of c) or the wavelength depending on the mode.
- Read the result, the full working line, and the derived panel that shows kinetic energy in joules and eV plus the regime label (atomic spacing, electron-microscopy, nuclear).
Everything runs in your browser. No data leaves your device.
Worked example
An electron is accelerated through a potential difference and reaches a speed of 1.00 × 10⁶ m/s. What is its de Broglie wavelength?
- m(electron) = 9.1094 × 10⁻³¹ kg
- v = 1.00 × 10⁶ m/s
- p = mv = 9.1094 × 10⁻³¹ × 1.00 × 10⁶ = 9.1094 × 10⁻²⁵ kg·m/s
- λ = h / p = 6.6261 × 10⁻³⁴ / 9.1094 × 10⁻²⁵ = 7.274 × 10⁻¹⁰ m ≈ 0.727 nm
This falls squarely in the Ångström regime (0.1–1 nm), comparable to the spacing between atoms in a crystal lattice — which is exactly why low-energy electrons diffract from crystal surfaces in LEED experiments.
The kinetic energy is KE = ½mv² = ½ × 9.109 × 10⁻³¹ × (10⁶)² ≈ 4.55 × 10⁻¹⁹ J ≈ 2.84 eV.
Formula note
The formula λ = h / (mv) is non-relativistic. It assumes the particle’s speed is well below the speed of light (c = 2.998 × 10⁸ m/s). As a rule of thumb, the correction becomes material above v ≈ 0.1c. At that speed the Lorentz factor γ ≈ 1.005 and the error is about 0.5 %. For higher speeds, replace mv with the relativistic momentum p = γmv where γ = 1 / sqrt(1 − v²/c²). The calculator shows a reminder when the velocity unit is set to fractions of c.
The constants used are the 2018 CODATA recommended values:
- h = 6.62607015 × 10⁻³⁴ J·s
- m_e = 9.1093837015 × 10⁻³¹ kg
- m_p = 1.67262192369 × 10⁻²⁷ kg
- m_n = 1.67492749804 × 10⁻²⁷ kg
- 1 u = 1.66053906660 × 10⁻²⁷ kg