What are the applications of delta function potentials?

Question

Are there real applications for using delta function potentials in quantum mechanics (other than using it as an exactly solvable toy model in introductory undergraduate quantum mechanics textbooks) ?

Answer 1

The best example I can think of is the modelling of a crystal via a series of equally spaced delta functions. This set of spaced delta functions is called a Delta comb and has several applications not only in quantum mechanics :-).

Answer 2

There are many applications, although you tend to see them disguised somewhat.

• The nonlinear Schrodinger equation in 1d: this is a quantum field theory describing gas of particles interacting with each other delta-function potentials. This is both theoretically important model, since it is solvable by Bethe-Ansatz, and experimentally used to model certain optical systems.
• The relativistic quantum $\phi^4$ model: in the nonrelativistic limit, this is a nonlinear Schrodinger equation, so it is the closest you can come to a relativistic gas of particles with delta-repulsion. The study of the N-copy version of this tells you the statistics of self-avoiding random walks in 2d and 3d, and this can be thought of as a path with an infinite delta-function repulsion to itself and to other paths of the same type. The analysis of this model, and it's connection to polymer self-avoidance is attributed to deGennes.

So the Higgs boson can be thought of as having a delta-function repulsion to other Higgs bosons in the standard model, or the closest relativistic analog. In addition to this, there is a simple universality result

• The scattering of any localized potential in 1d asymptotes to the scattering off a delta potential for long wavelengths.

This is also true in higher dimensions, if appropriately qualified. In 2d and above, there is an additional scale factor which tells you that the scattering is attenuated at long wavelengths. You can think of this as the probability of the random-walking path-integral finding the interaction region. The attenuation is by a factor which is analogous to the recurrence time of a random walk, it is logarithmic in |k| (for small |k|) 2d and by a power of k in higher dimensions. This means it is a useful toy model for renormalization.

For this reason, the delta-potential is 1-d specific. If you try to define a higher dimensional delta-potential, you need to renormalize the coefficient in the delta-function limit to get a fixed ground state energy, and really, in 3d and above, you don't have a sensible ground state. You can see this by doing the inverse problem--- start with a (real positive) ground state ansatz

$$ \psi_0(x) = e^{-W}$$

and find the potential which makes W a ground state:

$$ V(x) = {1\over 2} |\nabla W|^2 + {1\over 2} \nabla^2 W $$

In 1d, you can see that making $W=|x|$ gives the delta function (from the second term). In higher dimensions, you get the Coulomb force from the same ansatz. So the delta-well is a 1d analog of the Coulomb well in this way of thinking.

Even just for 1d, you can use the delta-well to describe a surface binding potential, since the motion in the perpendicular direction is bound. It is a very important model, since it is the universal point limit.