# 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.