# Schrödinger Equation for a One-Dimensional Delta Function Potential

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After the free particle, the most elementary example of a one-dimensional time-independent Schrödinger equation is conceptually that of a particle in a delta function potential: (in units with ). For an attractive potential, with , there is exactly one bound state, with and . Note that and . Since the delta function has dimensions of , this solution is considered the one-dimensional analog of a hydrogen-like atom. The bound state, in fact, resembles a cross section of a 1 orbital .

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: wavefunction for the single bound state

Snapshot 2: unperturbed continuum state, with

Snapshot 3: scattering state, showing incident, transmitted and reflected waves

Reference: S. M. Blinder, "Green's Function and Propagator for the One-Dimensional Delta Function Potential," *Phys. Rev. A*, 37(3), 1988 pp. 973–976.

## Permanent Citation