boundstate

A bound state is a quantum state in which a particle or system is confined by a potential in such a way that its energy lies below the threshold of the continuum. In practical terms, the particle remains localized instead of escaping to infinity, and the associated wavefunction is normalizable and decays at large distances.

Mathematically, bound states are solutions to the time-independent Schrödinger equation Hψ = Eψ with energy E less than the continuum threshold. For a particle in a potential V(x) that vanishes at infinity, bound-state energies satisfy E < 0 in the common convention, and the corresponding wavefunctions decrease exponentially as |x| → ∞. The energies of bound states are typically discrete, forming a finite or countably infinite set of eigenvalues, unlike scattering states whose energies form a continuum.

The binding energy is a measure of how strongly the state is bound. If E_threshold denotes the continuum onset, the binding energy is E_bind = E_threshold − E > 0. Bound states can arise in many systems, including the electron in a hydrogen atom, electrons in atoms and molecules, nucleons in nuclei, and excitons in semiconductors. More complex bound states include diatomic molecules, multi-electron atoms, and hadrons bound by strong interactions.

In addition to true bound states, systems may exhibit quasi-bound or resonant states with finite lifetimes, appearing as sharp features in scattering spectra. Bound-state concepts extend to multi-particle and relativistic contexts, where appropriate, and are central to understanding spectra, reaction dynamics, and stability across physics.