de Broglie Wavelength
The de Broglie wavelength is the wavelength associated with any moving particle, expressing the wave-like nature of matter as proposed by Louis de Broglie in 1924. It is inversely proportional to the momentum of the particle, meaning heavier or faster-moving objects have shorter wavelengths and thus exhibit negligible quantum wave behaviour. This concept was experimentally confirmed by electron diffraction experiments and forms the basis for electron microscopy and quantum confinement in nanomaterials.
Key Formula
λ = h / p = h / (m × v)
LaTeX: \lambda = \frac{h}{p} = \frac{h}{mv}
| Symbol | Meaning | Unit |
|---|---|---|
| λ | de Broglie wavelength | Metres (m) |
| h | Planck's constant (6.626 × 10⁻³⁴) | J·s |
| p | Momentum of the particle | kg·m/s |
| m | Mass of the particle | Kilograms (kg) |
| v | Velocity of the particle | m/s |
Worked Example
Problem
Calculate the de Broglie wavelength of an electron (mass = 9.11 × 10⁻³¹ kg) moving at 2.0 × 10⁶ m/s.
Solution
Step 1: Identify known values. m = 9.11 × 10⁻³¹ kg v = 2.0 × 10⁶ m/s h = 6.626 × 10⁻³⁴ J·s Step 2: Calculate momentum. p = mv = 9.11 × 10⁻³¹ × 2.0 × 10⁶ = 1.822 × 10⁻²⁴ kg·m/s Step 3: Apply de Broglie formula. λ = h / p = 6.626 × 10⁻³⁴ / 1.822 × 10⁻²⁴ Step 4: Compute. λ = 3.637 × 10⁻¹⁰ m = 0.364 nm
Answer
λ ≈ 3.64 × 10⁻¹⁰ m (0.364 nm), comparable to atomic spacing in crystals
de Broglie Wavelengths for Various Particles and Objects
| Object | Mass (kg) | Speed (m/s) | Wavelength (m) | Observable? |
|---|---|---|---|---|
| Electron | 9.11 × 10⁻³¹ | 2.0 × 10⁶ | 3.64 × 10⁻¹⁰ | Yes (X-ray scale) |
| Proton | 1.67 × 10⁻²⁷ | 2.0 × 10⁶ | 1.98 × 10⁻¹³ | Yes (nuclear scale) |
| Helium atom | 6.64 × 10⁻²⁷ | 1.0 × 10³ | 9.97 × 10⁻¹¹ | Yes (experiment) |
| Tennis ball | 0.057 | 30 | 3.88 × 10⁻³⁴ | No (negligible) |
| Car | 1500 | 30 | 1.47 × 10⁻³⁸ | No (negligible) |
Interactive Tools
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Related Terms
Wave-Particle Duality
Wave-particle duality is the quantum mechanical principle stating that every quantum entity, such as an electron or photon, exhibits both wave-like and particle-like properties depending on how it is observed or measured. In experiments such as the double-slit experiment, particles produce interference patterns characteristic of waves when not observed, but behave as localized particles when detected at specific positions. This duality is central to quantum mechanics and demonstrates that classical concepts of "wave" and "particle" are complementary rather than contradictory descriptions of quantum objects.
Photon
A photon is a massless elementary particle and the quantum of electromagnetic radiation, carrying energy proportional to its frequency. It is the force-carrier particle for the electromagnetic force and travels at the speed of light in a vacuum. Photons exhibit both wave and particle behaviour and are fundamental to understanding light-matter interactions, including the photoelectric effect, atomic emission spectra, and optical technologies.
Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that it is fundamentally impossible to simultaneously determine both the exact position and exact momentum of a quantum particle with arbitrary precision; the more precisely one is known, the less precisely the other can be known. This is not a limitation of measurement instruments but an intrinsic property of quantum systems arising from the wave nature of matter. A complementary relation exists between energy and time, and the principle has profound implications for atomic stability, electron orbitals, and the zero-point energy of quantum systems.
Named after French physicist Louis-Victor de Broglie (1892–1987), who proposed matter waves in his 1924 doctoral thesis. "Wavelength" is from Old English "wæglength", a compound of "wave" and "length". De Broglie received the Nobel Prize in Physics in 1929 for this discovery.