A quantum mechanics concept defining a model element, mathematical object, or experimental method used to predict measurable outcomes. It applies when required assumptions and definitions are specified and yields computable probabilities and expectation values. It does not ensure correctness without validation of approximations, numerical stability, and consistency of units and conventions. It materially affects interpretation of experiments and the reliability of theoretical predictions across quantum systems. The concept is generally stable, though methods and implementations evolve over time.

Principle

Bound states arise from solutions of the Schrödinger equation that satisfy square-integrability and boundary conditions leading to quantization of energy; mathematically they correspond to poles of the resolvent on the negative real axis (for typical reference) or discrete spectrum of the Hamiltonian.

Demonstration

Examples include the energy eigenstates of the infinite square well with discrete energies En ∝ n^2, the hydrogen atom's discrete negative-energy levels, or vibrational modes of a molecular potential well that localize the particle near equilibrium.

Misapplication

Labeling long-lived resonances or metastable states as true bound states without checking normalizability and spectral discreteness, or applying bound-state assumptions (discrete orthogonality, no current to infinity) to scattering or open systems.

Consequence

True bound states yield quantized energy spectra, stationary probability distributions, and selection rules for transitions; they underpin atomic and molecular spectroscopy and determine stability and binding energies of composite systems.

Reversal

The opposite concept is an unbound or scattering state: non-normalizable continuum eigenstates that extend to infinity, carry flux, and form the continuous spectrum rather than discrete, localized energy levels.

Boundary

Definition applies within the linear single-particle Hamiltonian framework (or separable effective single-particle channels); it excludes continuum resonances, virtual states, many-body correlated bound complexes requiring different criteria, and relativistic negative‑energy continua unless treated within appropriate frameworks.

Semantic Tension

There is tension between calling a state 'bound' because it is localized on experimental timescales (metastable) and the stricter mathematical notion of square-integrable, discrete-spectrum eigenstates; practical usage sometimes blurs this distinction.

Synthesis

A bound state is a square-integrable eigenstate of the Hamiltonian localized by a potential and associated with a discrete energy; it produces stable, quantized observables in contrast to continuum scattering states, though one must distinguish true mathematical boundness from long-lived metastability.