De Broglie Wavelength Calculator
Calculate the quantum mechanical wavelength of any particle from its mass and velocity or kinetic energy — revealing the wave-particle duality at the heart of quantum physics.
Enter the particle mass and either velocity or kinetic energy (or direct momentum) to calculate its De Broglie wavelength and related quantum properties.
De Broglie Wavelength Calculator
Calculate the quantum mechanical wavelength of any particle from its mass and velocity or kinetic energy — revealing the wave-particle duality at the heart of quantum physics.
About the De Broglie Wavelength Calculator
In 1924, French physicist Louis de Broglie made a revolutionary proposal: just as Einstein had shown that light (classically a wave) could behave as particles (photons), all particles of matter — electrons, protons, even everyday objects — should also exhibit wave-like properties. The wavelength associated with a moving particle is now called the De Broglie wavelength, and it is given by the elegant equation λ = h / p, where λ is the wavelength, h is Planck's constant (6.62607015×10⁻³⁴ J·s), and p is the particle's momentum.
Momentum can be expressed in several ways. For a particle with mass m moving at non-relativistic velocity v, p = mv, giving λ = h / (mv). If the particle's kinetic energy E is known instead, we use p = √(2mE), so λ = h / √(2mE). In some contexts the momentum is measured directly from experimental data such as particle track curvature in a magnetic field, in which case λ = h / p immediately. This calculator supports all three input modes.
The De Broglie wavelength decreases as momentum increases: faster particles or more massive particles have shorter wavelengths. For a 9.1×10⁻³¹ kg electron moving at 2.2×10⁶ m/s (typical for the ground state of hydrogen), the wavelength is about 0.33 nm — comparable to atomic bond lengths, which is why electrons diffract from crystal lattices and why electron microscopes can resolve individual atoms. By contrast, a 145 g baseball thrown at 40 m/s has a De Broglie wavelength of roughly 1.1×10⁻³⁴ m — many orders of magnitude smaller than any proton, explaining why quantum effects are utterly unobservable for macroscopic objects.
This wave nature of matter has profound practical consequences. Electron diffraction underpins transmission electron microscopy (TEM) and X-ray crystallography via Bragg's law. Quantum tunnelling — where a particle passes through a classically forbidden energy barrier — depends directly on the wavelength: longer wavelengths (lower momenta) tunnel more easily, which is why hydrogen nuclei can fuse in the sun at temperatures seemingly too low to overcome the Coulomb barrier. Neutron diffraction is used to determine crystal and molecular structures that are invisible to X-rays because neutrons scatter from atomic nuclei rather than electron clouds.
For relativistic particles where v approaches c, the non-relativistic p = mv underestimates momentum. The relativistic momentum is p = γmv = mv / √(1 − v²/c²). For electrons in a 1 MeV accelerator, relativistic corrections become significant. This calculator assumes non-relativistic speeds (v << c), which is valid for most laboratory applications except high-energy particle physics.
Worked Examples
Four representative cases spanning from subatomic particles to macroscopic objects.
| Particle / Scenario | De Broglie Wavelength | Significance |
|---|---|---|
| Electron in hydrogen atom ground state: m = 9.1094×10⁻³¹ kg, v = 2.2×10⁶ m/s | λ ≈ 3.31×10⁻¹⁰ m (0.331 nm) | Comparable to the Bohr radius. The electron's circumference in the ground state is exactly one wavelength, consistent with Bohr quantisation. |
| Proton in particle accelerator: m = 1.6726×10⁻²⁷ kg, KE = 1.6×10⁻¹² J | λ ≈ 9.06×10⁻¹⁵ m (0.00906 pm) | Deep sub-nuclear wavelength. At this energy, protons can probe the internal quark structure of other protons. |
| Thermal neutron: m = 1.6749×10⁻²⁷ kg, KE = 4.14×10⁻²¹ J (room temperature) | λ ≈ 1.78×10⁻¹⁰ m (0.178 nm) | Ideal for neutron diffraction. Wavelength matches typical interatomic spacings, making thermal neutrons perfect for crystal structure determination. |
| Baseball: m = 0.145 kg, v = 44.7 m/s (100 mph) | λ ≈ 1.02×10⁻³⁴ m | Wavelength is 10²⁰ times smaller than a proton. Quantum effects are completely negligible — classical physics applies perfectly. |
How to Use the De Broglie Wavelength Calculator
- Select the input mode: 'Mass + Velocity' if you know the particle's speed, 'Mass + Kinetic Energy' if you know its energy in joules, or 'Momentum (direct)' if you have measured the momentum directly.
- Enter the particle mass in kilograms. For common particles: electron = 9.1094×10⁻³¹ kg, proton = 1.6726×10⁻²⁷ kg, neutron = 1.6749×10⁻²⁷ kg. Convert g to kg by dividing by 1000.
- Enter the velocity in m/s, the kinetic energy in joules (multiply eV by 1.60218×10⁻¹⁹ to convert), or the momentum in kg·m/s, depending on your chosen input mode.
- Click Calculate. The results show the wavelength in metres, nanometres, and picometres, plus the momentum used and the corresponding frequency.
- Click Reset to clear the fields. Use the example buttons in the worked examples section to load representative particle data directly into the calculator.
Frequently Asked Questions
What is the De Broglie wavelength physically?
The De Broglie wavelength is the spatial period of the quantum mechanical wave function associated with a moving particle. It describes the scale over which quantum interference effects — such as diffraction and tunnelling — are significant. When this wavelength is comparable to the size of a system, quantum mechanics must be used; when it is vastly smaller than all relevant length scales, classical mechanics is sufficient.
How do I convert electron volts (eV) to joules?
Multiply by the elementary charge: 1 eV = 1.60218×10⁻¹⁹ J. For example, a 100 eV electron has kinetic energy 100 × 1.60218×10⁻¹⁹ = 1.60218×10⁻¹⁷ J. Enter this joules value in the Kinetic Energy field together with the electron mass to find the corresponding De Broglie wavelength.
Why does the calculator output wavelengths in nm and pm?
Nanometres (1 nm = 10⁻⁹ m) are convenient for electron wavelengths in the range 0.01–1 nm used in electron microscopy, and for UV and soft X-ray wavelengths. Picometres (1 pm = 10⁻¹² m) are used for X-ray crystallography and nuclear physics, where wavelengths are 1–100 pm. Metres is included as the SI base unit for completeness and for use in calculations.
Does this calculator account for relativistic effects?
No — the calculator uses non-relativistic momentum p = mv and p = √(2mE). This is accurate when the velocity is well below the speed of light. For electrons, relativistic corrections become significant above about 0.5 MeV (v > 0.86c). For protons and heavier particles, the threshold is proportionally higher. For extreme energies, use the relativistic momentum formula p = γmv.
What is the connection between De Broglie wavelength and electron microscopy?
The resolution of any microscope is limited to approximately half the wavelength of the probe. Visible light has wavelengths of 400–700 nm, limiting optical microscopes to roughly 200 nm resolution. Electrons accelerated to 100 keV have De Broglie wavelengths of about 0.004 nm — 50,000 times shorter — allowing transmission electron microscopes to image individual atoms with sub-Ångström resolution.
Can macroscopic objects truly have a De Broglie wavelength?
Yes, mathematically — but the wavelength is so astronomically small that it is physically undetectable. A 1 g marble moving at 1 m/s has λ ≈ 6.6×10⁻³¹ m, some 20 orders of magnitude smaller than a proton. No interference experiment could ever resolve such a wavelength with any foreseeable technology, which is why quantum effects are absent from everyday experience.