1D

1D denotes one-dimensionality, the property of having exactly one independent coordinate. In mathematics, a 1D object is a curve or line; examples include the real line and smooth curves parameterized by a single variable. A 1D manifold is locally like the real line, and the natural measure on it is length.

In physics, 1D systems are those whose dynamics effectively depend on a single coordinate, such as motion along a line. The classical wave equation on a string and the propagation of vibrations in a wire or optical fiber are standard 1D problems. The quantum mechanical description of a particle in a 1D potential uses the Schrödinger equation with one spatial coordinate, yielding discrete bound states in wells and characteristic scattering behavior.

Many-body and condensed-matter contexts describe electrons, atoms, or quasiparticles confined to quasi-one-dimensional environments—for example, quantum wires, carbon nanotubes, and ultracold atoms in elongated traps. In these systems interactions are enhanced by reduced dimensionality, leading to phenomena like strong quantum fluctuations and non-Fermi-liquid behavior. The low-energy physics is often captured by frameworks such as Luttinger liquid theory, with bosonization as a common analytical tool.

From a statistical-mechanics perspective, many short-range 1D models do not exhibit a finite-temperature phase transition; the one-dimensional Ising model, for instance, has no phase transition at any finite temperature. Nonetheless, 1D systems remain fundamental as exactly solvable models and as simplified arenas to study spectra, correlations, and critical phenomena before generalizing to higher dimensions.