De Broglie Wavelength Calculator
Calculate the de Broglie wavelength, mass, velocity, or momentum of any particle using λ = h/(mv) where h = 6.62607015×10⁻³⁴ J·s.
Related: | | |
What is de Broglie Wavelength?
The de Broglie wavelength is the quantum wavelength associated with every moving particle. Proposed by Louis de Broglie in 1924, the wavelength λ is determined by the particle’s momentum: a particle with greater momentum has a shorter wavelength. The fundamental relationship is:
λ = h / p = h / (m × v)
Where λ is the wavelength in meters, h = 6.62607015×10⁻³⁴ J·s is Planck’s constant, p is momentum (kg·m/s), m is mass (kg), and v is velocity (m/s). This equation is one of the cornerstones of quantum mechanics, establishing that all matter has both particle and wave properties.
How to Use This Calculator
Select the calculation mode from the tabs above. Enter the known values and select appropriate units. The calculator automatically computes the unknown quantity using the de Broglie relation. For example, to find an electron’s wavelength, enter the electron mass (9.109×10⁻³¹ kg) and velocity, and the calculator returns the wavelength in your selected unit.
Reference de Broglie Wavelengths
| Particle / Object | Mass (kg) | Velocity (m/s) | Wavelength |
|---|---|---|---|
| Electron (1 eV kinetic energy) | 9.109e-31 | 5.930e+5 | 1.23 nm |
| Electron (100 eV kinetic energy) | 9.109e-31 | 5.930e+6 | 0.123 nm |
| Electron (10 keV kinetic energy) | 9.109e-31 | 5.930e+7 | 12.3 pm |
| Neutron (thermal, 0.025 eV) | 1.675e-27 | 2.187e+3 | 0.18 nm |
| Proton (1 MeV) | 1.673e-27 | 1.384e+7 | 28.7 fm |
| Baseball (40 m/s) | 1.450e-1 | 4.000e+1 | 1.14×10⁻³⁴ m |
| Dust particle (1 μm, 1 m/s) | 1.000e-12 | 1.000e+0 | 6.63×10⁻²² m |
| Electron in H atom (Bohr orbit) | 9.109e-31 | 2.180e+6 | 0.333 nm |
Wave-Particle Duality
Wave-particle duality is the concept that every quantum entity exhibits both wave and particle properties. The de Broglie wavelength quantifies the wave aspect: longer wavelengths mean more pronounced wave behavior. This duality is demonstrated clearly in the double-slit experiment, where electrons, neutrons, and even large molecules produce interference patterns characteristic of waves, yet are detected as individual particles.
The wave nature of matter is not observable at macroscopic scales because the de Broglie wavelength is inversely proportional to momentum. An object with a mass of 1 gram moving at 1 m/s has a wavelength of about 6.6×10⁻³¹ m, far too small to detect. However, for an electron, the wavelength is comparable to atomic dimensions, making quantum effects prominent.
Related Calculators
Authoritative References
Frequently Asked Questions
What is de Broglie wavelength?
The de Broglie wavelength (λ) is the quantum mechanical wavelength associated with any moving particle. It is defined as λ = h/p, where h is Planck's constant and p is the particle's momentum. This concept introduced wave-particle duality, showing that all matter exhibits both particle and wave properties.
What is the de Broglie equation?
The de Broglie equation is λ = h/p = h/(m×v), where λ is the de Broglie wavelength, h = 6.62607015×10⁻³⁴ J·s is Planck's constant, p is momentum, m is mass, and v is velocity. This equation forms the foundation of wave mechanics.
Who proposed matter waves?
Louis de Broglie proposed matter waves in his 1924 PhD thesis, for which he won the Nobel Prize in Physics in 1929. He suggested that all particles have an associated wavelength, fundamentally establishing the wave-particle duality concept. His idea was revolutionary and led to the development of quantum mechanics.
How is de Broglie wavelength calculated?
De Broglie wavelength is calculated using λ = h/(m×v). First determine the particle's mass in kilograms and velocity in meters per second. Multiply mass by velocity to get momentum, then divide Planck's constant (6.62607015×10⁻³⁴) by that momentum. The result is the wavelength in meters.
What is the significance of de Broglie wavelength?
The de Broglie wavelength is significant because it unified particle and wave descriptions of nature. It explains why quantization occurs in atomic systems (standing matter waves), enables technologies like electron microscopy and neutron scattering, and provides the foundation for quantum mechanical wavefunctions.
Why do macroscopic objects not show wave behavior?
Macroscopic objects have extremely small de Broglie wavelengths due to their large mass. For example, a baseball moving at 40 m/s has a wavelength of about 10⁻³⁴ m, far smaller than any measurable scale. The wavelength is inversely proportional to mass, so everyday objects have undetectably tiny wave properties.
What is the de Broglie wavelength of an electron?
An electron accelerated through 1 volt has a de Broglie wavelength of about 1.23 nm. At 100 V it is about 0.123 nm, and at 10 kV it is about 0.0123 nm. This wavelength range makes electrons ideal for microscopy, as their wavelength is comparable to atomic spacings in crystals.
What particles exhibit wave-particle duality?
All particles exhibit wave-particle duality, including electrons, protons, neutrons, atoms, molecules, and even buckminsterfullerene (C₆₀) molecules. Massive particles have proportionally smaller wavelengths. Photons also exhibit duality but their wavelength is given by λ = c/ν rather than the de Broglie relation.
How does the double-slit experiment demonstrate matter waves?
The double-slit experiment with electrons directly demonstrates matter waves. When electrons are fired one at a time through two slits, they produce an interference pattern on the detector, exactly like light waves. This pattern disappears when which-path information is observed, proving the wave nature of particles.
What is the relationship between momentum and wavelength?
De Broglie wavelength is inversely proportional to momentum: λ = h/p. Higher momentum means shorter wavelength, and lower momentum means longer wavelength. Since momentum equals mass times velocity, increasing either mass or velocity decreases the wavelength proportionally.
How does mass affect de Broglie wavelength?
Mass inversely affects de Broglie wavelength — larger mass produces smaller wavelength at the same velocity. This is why we can observe quantum effects with electrons but not with baseballs. The wavelength scales as 1/m, so doubling mass halves the wavelength.
What is the de Broglie wavelength of a proton?
A proton's de Broglie wavelength depends on its kinetic energy. At 1 MeV, a proton has a wavelength of about 28.7 fm (femtometers). At higher energies, the wavelength becomes even shorter, which is why protons are used to probe nuclear structure in particle accelerators.
Can we observe matter waves directly?
Matter waves cannot be directly observed in the same way as water waves or light waves. Their effects are observed indirectly through interference and diffraction patterns. Electron diffraction from crystals provides clear evidence of matter waves, as seen in Davisson-Germer experiments.
What is the role of Planck’s constant in de Broglie wavelength?
Planck's constant (h = 6.62607015×10⁻³⁴ J·s) is the fundamental proportionality constant linking momentum to wavelength. It defines the scale at which quantum effects become significant. The tiny value of h explains why wave behavior is only noticeable for microscopic particles.
How does velocity affect de Broglie wavelength?
De Broglie wavelength is inversely proportional to velocity. Doubling velocity halves the wavelength. At non-relativistic speeds, wavelength decreases linearly with increasing speed. At relativistic speeds, the relativistic momentum (p = γmv) must be used instead of classical momentum.
What instruments use de Broglie wavelength?
The most important instrument is the electron microscope, which uses the short wavelength of electrons (about 100,000× shorter than visible light) to achieve nanometer resolution. Neutron diffractometers use neutron de Broglie wavelengths to study crystal structures and magnetic materials.
What is an electron microscope and how does it use de Broglie wavelength?
An electron microscope uses a beam of electrons accelerated to high velocity, producing a de Broglie wavelength thousands of times shorter than visible light. This enables imaging at the atomic scale. Transmission electron microscopes (TEM) can achieve resolution below 0.1 nm, revealing individual atoms.
What is the de Broglie wavelength of a neutron?
A thermal neutron at room temperature (0.025 eV) has a de Broglie wavelength of about 0.18 nm, comparable to atomic spacings in solids. This makes neutrons ideal for studying crystal structures through neutron diffraction. Cold neutrons can have wavelengths exceeding 1 nm.
What is thermal de Broglie wavelength?
The thermal de Broglie wavelength is λ = h/√(2πmkT), representing the average de Broglie wavelength of particles in a gas at temperature T. It sets the quantum length scale for particle interactions and is used in statistical mechanics to determine whether quantum effects are important (when λ exceeds interparticle spacing).
How is de Broglie wavelength related to quantum mechanics?
De Broglie's hypothesis was the starting point for Schrödinger’s development of wave mechanics. The Schrödinger equation describes how matter waves evolve in space and time. The wavelength determines the spatial oscillations of the quantum wavefunction, directly connecting particle momentum to the wavefunction’s phase.
What is the Compton wavelength vs de Broglie wavelength?
The Compton wavelength (λc = h/mc) is a fundamental length scale for a particle, related to quantum field theory and particle creation. The de Broglie wavelength (λ = h/p) depends on momentum and varies with velocity. For a particle at rest, the Compton wavelength is the limiting de Broglie wavelength at v = c.
Does a photon have a de Broglie wavelength?
A photon’s wavelength is given by λ = c/ν, which is equivalent to the de Broglie relation since photon momentum is p = h/λ. The de Broglie relation λ = h/p thus applies universally to all particles, including photons. For photons, p = E/c = h u/c.
What is the de Broglie-Bohm theory?
The de Broglie-Bohm theory (also called pilot-wave theory) is an interpretation of quantum mechanics where particles have definite positions guided by a pilot wave. De Broglie presented this at the 1927 Solvay Conference. It reproduces all standard quantum predictions while providing a deterministic hidden-variables description.
How do you convert between wavelength units?
To convert between wavelength units: 1 m = 100 cm = 1000 mm = 10⁶ μm = 10⁹ nm = 10¹² pm = 10¹⁵ fm. For example, 1 nm = 10⁻⁹ m. Electron wavelengths are typically in nm or pm, while nuclear-scale wavelengths are in fm.
What is the de Broglie wavelength of a baseball?
A standard baseball (0.145 kg) thrown at 40 m/s has a de Broglie wavelength of about 1.14×10⁻³⁴ m. This is roughly 10²⁰ times smaller than the size of a proton, making quantum effects completely negligible. This illustrates why everyday objects behave classically.
Why is de Broglie wavelength important in chemistry?
De Broglie wavelengths explain atomic orbitals, which are standing matter waves around the nucleus. The quantization of electron orbits arises when the de Broglie wavelength fits around the nucleus an integer number of times. This determines chemical bonding, molecular structure, and periodic properties.
What is electron diffraction?
Electron diffraction occurs when electrons scatter from a crystal lattice, producing interference patterns due to their wave nature. The Davisson-Germer experiment (1927) first demonstrated electron diffraction from nickel crystals. This technique is now used to determine surface structures and in transmission electron microscopy.
How are matter waves used in neutron scattering?
Neutron scattering uses the de Broglie wavelength of neutrons to probe material structure. Thermal neutrons with wavelengths around 0.1 nm are ideal for studying atomic positions and magnetic moments. Longer-wavelength cold neutrons investigate larger-scale structures like polymers and biological molecules.
What is the relationship between de Broglie wavelength and kinetic energy?
For non-relativistic particles, λ = h/√(2mE), where E is kinetic energy. This is because p = √(2mE). So wavelength is inversely proportional to the square root of kinetic energy, mass, or both. For relativistic particles, the full relativistic momentum must be used: p = √(E² - m²c⁴)/c.
Does temperature affect de Broglie wavelength?
Yes, temperature affects de Broglie wavelength through particle velocity. In a gas at thermal equilibrium, particles have a Maxwell-Boltzmann velocity distribution. The average de Broglie wavelength decreases with increasing temperature as λ ∝ 1/√T. At very low temperatures, wavelengths become large enough for quantum effects like Bose-Einstein condensation.