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.
Single delta potential
The time-independent Schrödinger equation for the wave function of a particle in one dimension in a potential is
where is the reduced Planck constant, and is the energy of the particle.
The delta potential is the potential
where is the Dirac delta function.
It is called a delta potential well if is negative, and a delta potential barrier if 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.
Solving the Schrödinger equation
The potential splits the space in two parts ( and ). 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 and , where the wave number 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 ), represents a wave traveling to the right, and one traveling to the left.
One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:
A second relation can be found by studying the derivative of the wavefunction. Normally, we could also impose differentiability at the origin, but this is not possible because of the delta potential. However, if we integrate the Schrödinger equation around , over an interval :
In the limit as , the right-hand side of this equation vanishes; the left-hand side becomes
because
Substituting the definition of into this expression yields
The boundary conditions thus give the following restrictions on the coefficients
Bound state (E < 0)
In any one-dimensional attractive potential there will be a bound state. To find its energy, note that for , is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of x (see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms: . The wave function is then
From the boundary conditions and normalization conditions, it follows that
from which it follows that must be negative, that is, the bound state only exists for the well, and not for the barrier. The Fourier transform of this wave function is a Lorentzian function.
The energy of the bound state is then
Scattering (E > 0)
For positive energies, the particle is free to move in either half-space: or . It may be scattered at the delta-function potential.
The quantum case can be studied in the following situation: a particle incident on the barrier from the left side . It may be reflected or transmitted . To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations (incoming particle), (reflection), (no incoming particle from the right) and (transmission), and solve for and even though we do not have any equations in . The result is
Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. The result is that there is a non-zero probability