Delta potential

Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value.

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.

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

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:

One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:

To find its energy, note that for E < 0, k = i√2m|E|/ħ = iκ 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: Ar = Bl = 0.

The quantum case can be studied in the following situation: a particle incident on the barrier from the left side (Ar).

To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations Ar = 1 (incoming particle), Al = r (reflection), Bl = 0 (no incoming particle from the right) and Br = t (transmission), and solve for r and t even though we do not have any equations in t. 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.

This does not depend on the sign of λ, that is, a barrier has the same probability of reflecting the particle as a well.

In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an effective mass m. Often, the surfaces of such materials are covered with oxide layers or are not ideal for other reasons.

This thin, non-conducting layer may then be modeled by a local delta-function potential as above.

In that case, the barrier is due to the air between the tip of the STM and the underlying object.

For a more general model of this situation, see Finite potential barrier (QM).

On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others.

The Schrödinger equation may then be reduced to the case considered here by an Ansatz for the wave function of the type

Alternatively, it is possible to generalize the delta function to exist on the surface of some domain D (see Laplacian of the indicator).

[2] The delta function model is actually a one-dimensional version of the Hydrogen atom according to the dimensional scaling method developed by the group of Dudley R. Herschbach[3] The delta function model becomes particularly useful with the double-well Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section.

The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:

is the "internuclear" distance with Dirac delta-function (negative) peaks located at x = ±R/2 (shown in brown in the diagram).

Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set

Matching of the wavefunction at the Dirac delta-function peaks yields the determinant

The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where A = B, and is called gerade.

Correspondingly, the "−" case is the wave function that is anti-symmetric about the midpoint, where A = −B, and is called ungerade (shown in green in the diagram).

They represent an approximation of the two lowest discrete energy states of the three-dimensional

Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by[4]

In the case of unequal charges, and for that matter the three-dimensional molecular problem, the solutions are given by a generalization of the Lambert W function (see Lambert W function § Generalizations).

For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the transmission coefficient is unity at zero energy.