Formula & Calculator
de Broglie Wavelength
Calculates the wavelength associated with any moving particle based on its momentum, reflecting the wave-particle duality of matter.
Interpretation
de Broglie wavelength: λ = h/p, where h is Planck's constant, p is momentum. It shows wave‑particle duality: particles have wavelength. Example: Electron with momentum 1e-24 kg·m/s → λ = 6.63e-34/1e-24 = 6.63e-10 m.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| lambda | de Broglie wavelength | m |
| h | Planck's constant | 6.626e-34 J.s |
| p | Momentum of the particle | kg.m/s |
What it means
The de Broglie wavelength is the wavelength associated with a moving particle, demonstrating that matter has wave‑like properties. The formula λ = h/p relates the wavelength to momentum. This concept is central to quantum mechanics and explains phenomena like electron diffraction and the quantization of atomic orbits. It is used in electron microscopy, where electrons are used to image objects at atomic scale. The de Broglie wavelength also appears in the Heisenberg uncertainty principle and in the development of wave mechanics. Understanding this duality is fundamental to modern physics.
Worked example
de Broglie Wavelength – Two Examples
Real‑World| Parameter | Value |
|---|---|
| p | 1×10⁻²⁴ kg·m/s |
| Parameter | Value |
|---|---|
| p | 5×10⁻²⁴ |
Common mistakes
- Momentum p: For a particle – p = mv (non‑relativistic) or relativistic.
- de Broglie wavelength λ: In metres – typically very small for macroscopic objects.
- Planck’s constant h: 6.626×10⁻³⁴ J·s.
- Wave‑particle duality: Applicable to all matter, not just subatomic particles.
- Units: p in kg·m/s → λ in m.
Applications
The de Broglie wavelength, λ = h/p, associates a wavelength with a particle, highlighting the wave‑particle duality. This concept is essential in quantum mechanics and is used in electron microscopy, neutron diffraction, and particle physics. Engineers apply it in the design of electron beam lithography, in scanning tunnelling microscopes, and in materials characterisation. The wavelength of electrons determines the resolution in electron microscopes. By understanding de Broglie's relation, professionals can exploit wave properties of matter to achieve high‑resolution imaging and to study the structure of matter at the atomic level.
- Design of electron microscopes (TEM, SEM)
- Electron beam lithography for semiconductor fabrication
- Neutron diffraction for crystal structure analysis
- Scanning tunnelling microscopy (STM)
- Quantum interference and atom optics
Frequently Asked Questions
The de Broglie wavelength is the wavelength associated with a moving particle, given by λ = h/p, where h is Planck's constant and p is the momentum (p = mv for non‑relativistic particles). It reflects the wave‑particle duality of matter.
Applying it to macroscopic objects (e.g., a baseball) where the wavelength is incredibly small and undetectable. The wave properties are only observable for microscopic particles (e.g., electrons).
The electron gains kinetic energy 1 eV = 1.6×10⁻¹⁹ J. Its momentum is p = √(2mE). Using h, λ ≈ 1.23 nm. This is comparable to atomic spacings.
Electron microscopes use the wave nature of electrons to achieve much higher resolution than optical microscopes, because the de Broglie wavelength of electrons is much smaller than visible light.
As speed increases, momentum increases, so wavelength decreases.
For particles moving close to the speed of light, use λ = h / (γ·m·v), where γ is the Lorentz factor.
Bohr proposed that electrons orbit the nucleus only in circular orbits where the circumference equals an integer multiple of the de Broglie wavelength: 2πr = nλ. This quantises angular momentum.
The Davisson‑Germer experiment (electron diffraction) and the Thomson experiment confirmed the wave nature of electrons.