154 Wave-Particle Duality
A principle that is used to treat matter and light as both particles and waves, where a particle is a localized quantum of energy and momentum, and where a wave is an extended oscillation that can interfere.
The de Broglie relation. A material particle of momentum \(p\) has a wavelength \(h/p\). This principle is used to assign a wave to an electron, a neutron, or any other massive particle.
The de Broglie wavelength is
\[ \lambda = \dfrac{h}{p} \]
where
- \(\lambda\) is the de Broglie wavelength.
- \(h\) is Planck’s constant.
- \(p\) is the momentum of the particle.
Photon momentum. A photon of wavelength \(\lambda\) has momentum \(h/\lambda\). This principle is used to assign a particle momentum to a light quantum.
The photon momentum is
\[ p = \dfrac{h}{\lambda} \]
where
- \(p\) is the photon momentum.
- \(h\) is Planck’s constant.
- \(\lambda\) is the wavelength.
The Planck relation. A photon of frequency \(f\) has energy \(hf\). This principle is used to treat light as discrete energy packets.
The Planck relation is
\[ E = hf \]
where
- \(E\) is the energy of one photon.
- \(h\) is Planck’s constant.
- \(f\) is the frequency.
The condition for observable matter waves. Interference and diffraction appear whenever the de Broglie wavelength is comparable to the size of an obstacle or slit. This principle is used to predict when electrons and other particles show wave behavior.
154.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — de Broglie wavelength and photon energy.
- Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §1.1 — wave-particle duality.
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