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Concept
Delta potential
Summary
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.
This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.
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 following results.
The potential splits the space in two parts (x < 0 and x > 0). In each of these parts the potential is zero, and the Schrödinger equation reduces to
this is a linear differential equation with constant coefficients, whose solutions are linear combinations of eikx and e−ikx, where the wave number k is related to the energy by
In general, due to the presence of the delta potential in the origin, the coefficients of the solution need not be the same in both half-spaces:
where, in the case of positive energies (real k), eikx represents a wave traveling to the right, and e−ikx one traveling to the left.
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