AcademyPhotons
Academy
Wave-Particle Duality
Level 1 - Physics topic page in Photons.
Principle
Quantum objects show both wave-like and particle-like behavior. Light travels and interferes as a wave, but exchanges energy and momentum in discrete photon packets.
Matter also has wave behavior, described by the de Broglie wavelength.
Notation
\(E\)
particle or photon energy
\(\mathrm{J,\;eV}\)
\(p\)
momentum magnitude
\(\mathrm{kg\,m\,s^{-1}}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(f\)
frequency
\(\mathrm{s^{-1}}\)
\(h\)
Planck constant
\(\mathrm{J\,s}\)
\(K\)
non-relativistic kinetic energy
\(\mathrm{J}\)
Method
Derivation 1: Photon relations
For light, wave variables and particle variables are linked by Planck's constant.
Photon energy
\[E=hf\]
Photon momentum
\[p=\frac{h}{\lambda}\]
Derivation 2: Matter waves
A particle with momentum \(p\) has de Broglie wavelength.
de Broglie wavelength
\[\lambda=\frac{h}{p}\]
Non-relativistic momentum
\[p=\sqrt{2mK}\]
Derivation 3: Interpret experiments
Interference and diffraction reveal wave behavior. Localized detection and energy transfer reveal particle behavior.
Photoelectric photon energy
\[K_{\max}=hf-\phi\]
Diffraction scale
\[a\sin\theta\sim\lambda\]
Rules
Photon energy
\[E=hf\]
Photon momentum
\[p=\frac{h}{\lambda}\]
de Broglie wavelength
\[\lambda=\frac{h}{p}\]
Non-relativistic momentum
\[p=\sqrt{2mK}\]
Examples
Question
Find the photon momentum for
\[\lambda=500\,\mathrm{nm}\]
Answer
\[p=\frac{h}{\lambda}=\frac{6.63\times10^{-34}}{500\times10^{-9}}=1.33\times10^{-27}\,\mathrm{kg\,m\,s^{-1}}\]
Checks
- Wave-particle duality does not mean a classical wave plus a classical particle at the same time.
- The observed behavior depends on the measurement setup.
- Interference can build up one particle at a time.
- A smaller momentum means a larger de Broglie wavelength.