Schrödinger Equation for a One-Dimensional Delta Function Potential

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 .
For , free particles are scattered by a delta function potential. The positive-energy solutions can be written , with . The amplitudes of the transmitted and reflected waves are accordingly given by and , respectively. Note that these are the same for attractive and repulsive delta funtion potentials, independent of the sign of .
For continuum states, the graphic shows a wave incident from the left. The transmitted wave is shown on the right in blue and the reflected wave, on the left in red, with opacities indicating relative wave amplitudes.



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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.
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