Single Delta Potential
The time-independent Schrödinger equation for the wave function ψ(x) of a particle in one dimension in a potential V(x) is
where ħ is the reduced Planck constant and E is the energy of the particle.
The delta potential is the potential
where δ(x) is the Dirac delta function. It is called a delta potential well if λ is negative and a delta potential barrier if λ is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the proceeding results.
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