http://demonstrations.wolfram.com/SchroedingerEquationForAOneDimensionalDeltaFunctionPotential/
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 
.
(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 [more]
, 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 [more]
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