 Delta potential

A one-dimensional delta potential is the point interaction $V(x)=g\delta (x)$, where $g$ has energy times length units. Rather than multiplying ordinary functions at a singular point, one can define it by matching a continuous wavefunction with a derivative jump:Integrating the Schrödinger equation across the origin gives this condition. It provides an exactly solvable model of a very narrow potential barrier for $g>0$, or a narrow attractive well for $g<0$. Scattering states are plane waves away from the origin, while the attractive interaction has a single bound state. The formula is specific to one dimension; a naive higher-dimensional Dirac delta function interaction needs additional care.

Delta potential, often referred to as the Dirac delta potential, is a mathematical construct used in quantum mechanics and quantum field theory. It represents an idealized potential energy function that is localized at a single point in space. The Dirac delta function, denoted as \(\delta(x - x_0)\), is defined such that: 1. \(\delta(x - x_0) = 0\) for all \(x \neq x_0\), 2.