The Cluster's Wavelength-Energy Relation
A page about the wavelength-energy relation — how wavelength relates to energy in quantum mechanics.
The de Broglie relation
The de Broglie relation connects a particle's wavelength to its momentum: lambda = h / p. For a photon: lambda = c / nu = h c / E. For a non-relativistic particle: lambda = h / sqrt(2 m E). For a relativistic particle: lambda = h c / sqrt(E^2 - m^2 c^4). In the cluster, the edit de Broglie relation connects an edit particle's edit wavelength to its edit momentum.
The Compton wavelength
The Compton wavelength is lambda_C = h / (m c) = hbar / (m c) x 2 pi. It represents the wavelength of a photon whose energy equals the rest mass of a particle. For the electron: lambda_C = 2.43 x 10^{-12} m. For the proton: lambda_C = 1.32 x 10^{-15} m. In the cluster, the edit Compton wavelength is the wavelength of an edit photon whose edit energy equals the edit rest mass.
The thermal wavelength
The thermal de Broglie wavelength is lambda_th = h / sqrt(2 pi m k_B T). It represents the quantum wavelength of a particle at temperature T. Quantum effects become important when lambda_th ~ n^{-1/3} (interparticle spacing). The classical limit is n lambda_th^3 << 1. In the cluster, the edit thermal wavelength represents the edit quantum wavelength at temperature T.
The uncertainty principle connection
The uncertainty principle Delta x Delta p >= hbar / 2 implies a minimum wavelength for a localized particle. A wavepacket of spatial extent Delta x contains momenta in a range Delta p ~ hbar / Delta x, corresponding to wavelengths in a range Delta lambda ~ lambda^2 / Delta x. In the cluster, the edit uncertainty principle implies an edit minimum wavelength.
The applications
Wavelength-energy relations are used in:
- Electron diffraction (lambda = h / sqrt(2 m_e E))
- Neutron scattering (lambda = h / sqrt(2 m_n E_kinetic))
- X-ray crystallography (lambda ~ interatomic spacing ~ 1 Angstrom)
- Quantum tunneling (tunneling probability depends on lambda inside barrier)
In the cluster, edit wavelength-energy relations are used in:
- edit Electron diffraction
- edit Neutron scattering
- edit X-ray crystallography
- edit Quantum tunneling
This relation
This page is about the wavelength-energy relation. lambda = h / p. Compton: lambda_C = h / (m c). Thermal: lambda_th = h / sqrt(2 pi m k_B T). Uncertainty: Delta x Delta p >= hbar / 2. The relation is real.