Calculate the de Broglie wavelength λ = h/p for any particle. Relativistic support, multiple input modes, energy scans, and scale comparisons.
Every particle with momentum has an associated wavelength, as proposed by Louis de Broglie in 1924: λ = h/p, where h is Planck's constant and p is the momentum. This wave nature of matter is directly observable in electron diffraction, neutron scattering, and even diffraction of large molecules like C60 fullerene.
For non-relativistic particles, λ = h/√(2mE), where m is mass and E is kinetic energy. For an electron accelerated through 100V, λ = 0.123 nm — comparable to atomic spacings, which is why electron microscopes can resolve atomic structures. For everyday objects, the wavelength is so tiny (~10⁻³⁴ m for a thrown baseball) that wave effects are undetectable.
This calculator computes the de Broglie wavelength from kinetic energy, velocity, momentum, or accelerating voltage, with full relativistic corrections. It supports electrons, protons, neutrons, alpha particles, muons, and custom particles, with an energy scan table and logarithmic scale comparison. Check the example with realistic values before reporting.
Computing de Broglie wavelengths involves fundamental constants, unit conversions between eV and joules, and (for high energies) relativistic momentum calculations. Comparing wavelengths to physical length scales helps determine when quantum effects are important.
This calculator handles all unit conversions and relativistic corrections automatically, supports four input modes (energy, velocity, momentum, voltage), and provides energy scan tables for quick reference. It is useful for students, researchers, and engineers working with electron microscopy, neutron scattering, or quantum mechanics.
λ = h/p. Non-relativistic: λ = h/√(2mKE). Relativistic: p = √(KE² + 2KE·mc²)/c. Reduced: ƛ = ℏ/p = λ/(2π). From voltage: λ = h/√(2meV).
Result: λ = 0.1226 nm (1.226 Å)
p = √(2 × 9.109×10⁻³¹ × 100 × 1.602×10⁻¹⁹) = 5.403×10⁻²⁴ kg·m/s. λ = 6.626×10⁻³⁴ / 5.403×10⁻²⁴ = 1.226×10⁻¹⁰ m = 0.1226 nm. This is comparable to atomic bond lengths.
In 1924, Louis de Broglie proposed that particles have an associated wavelength λ = h/mv, extending Einstein's photon concept to all matter. This audacious idea was confirmed in 1927 by Davisson and Germer, who observed electron diffraction from a nickel crystal, and independently by G.P. Thomson using thin gold foils. Both experiments showed diffraction patterns consistent with the de Broglie wavelength.
The discovery had profound implications: if electrons have wave properties, they should have standing-wave solutions in atoms — directly explaining Bohr's quantization condition. Schrödinger developed his wave equation (1926) specifically to describe these matter waves, leading to modern quantum mechanics.
The scanning electron microscope (SEM) and transmission electron microscope (TEM) exploit the short de Broglie wavelength of high-energy electrons to image structures far below the optical diffraction limit. At 200 keV, λ = 2.5 pm — theoretically allowing sub-angstrom resolution.
In practice, lens aberrations limit SEM resolution to ~1 nm and TEM to ~0.5 Å (50 pm). Modern aberration-corrected TEMs achieve ~0.4 Å, routinely imaging individual atoms in crystals. Cryo-EM, which won the 2017 Nobel Prize in Chemistry, uses low-dose electron beams to determine 3D structures of biological molecules at near-atomic resolution.
The frontier of matter wave physics is testing quantum superposition with increasingly large and complex objects. Interference has been demonstrated for molecules containing up to ~2000 atoms and masses of ~25,000 amu. These experiments constrain theories that propose quantum mechanics breaks down above some mass scale (objective collapse models). Current experiments push toward the 10⁶ amu range, using advanced gratings and precise vibration isolation.
All matter exhibits both particle and wave properties. The de Broglie wavelength describes the wave aspect: λ = h/p. For macroscopic objects, λ is so small as to be undetectable. For subatomic particles at low energies, λ is comparable to atomic dimensions and wave effects (diffraction, interference) are prominent.
When KE is a significant fraction of rest mass energy (mc²). For electrons, mc² = 0.511 MeV, so corrections matter above ~50 keV. For protons, mc² = 938 MeV, so non-relativistic formulas work up to ~100 MeV. The calculator shows the error from using non-relativistic formulas.
At 200 keV (typical TEM), electrons have λ ≈ 2.5 pm — 100,000× shorter than visible light. This allows resolving individual atoms. The resolution limit is approximately 0.61λ/NA, so shorter wavelength means finer detail.
Photons also satisfy λ = h/p, but since photons are massless, p = E/c = hf/c = h/λ, giving the familiar λ = c/f. For photons, the de Broglie and electromagnetic wavelengths are identical.
Yes — in 1999, diffraction of C60 fullerene (720 amu) was demonstrated. Later experiments showed interference patterns for molecules up to ~25,000 amu. The wavelength is extremely short (sub-picometer) but detectable with sensitive instruments.
At room temperature (~0.025 eV), neutrons have λ ≈ 0.18 nm — comparable to crystal lattice spacings. This makes thermal neutrons ideal for diffraction studies of crystal structure, complementary to X-ray diffraction.