Delta potential

A delta potential is an idealized interaction concentrated at a single point and represented by a Dirac delta function. In quantum mechanics, it models a sharply localized barrier or well.

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Dirac delta function: A distribution that integrates a test function by evaluating it at a single point. The potential is defined by multiplying this distribution by a strength parameter.
Bound state: A quantum state localized by an attractive interaction and having energy below the continuum threshold. An attractive delta potential supports exactly one bound state in one dimension.
Distribution theory: A mathematical framework that treats generalized functions through their action on test functions. It gives a rigorous meaning to a potential too singular to be an ordinary function.
Finite square well: A potential well with constant negative potential over a finite interval and zero potential outside. Unlike a delta well, it has nonzero width and can support multiple bound states.
Schrödinger equation: The fundamental wave equation governing nonrelativistic quantum systems. Solving it across the point interaction determines the wavefunction and its derivative jump.
Quantum scattering: The study of how quantum waves change when they encounter an interaction or potential. The delta potential provides an exactly solvable example of reflection and transmission.
Potential energy: Energy associated with an object's position or configuration in a force field. A delta potential is a singular spatial profile for potential energy.
Infinite potential barrier: An idealized region of infinite potential energy that excludes a particle from that region. A delta barrier has finite integrated strength, so particles can tunnel through it.
Boundary conditions: Constraints specifying how a solution or its derivatives behave at a domain's boundaries or interfaces. The delta spike leaves the wavefunction continuous but makes its derivative discontinuous.
Reflection coefficient: The fraction of incident wave intensity reflected by a scattering interaction. Its energy dependence quantifies how strongly a delta barrier reflects particles.