# History of delta barrier in quantum mechanics

I'm interested in finding something out about the history of the problem of the delta potential barrier in quantum mechanics.

Which was the first study to propose this problem, and perhaps any particular motivation for it?

I am aware of some of the usual motivations for using it, but any examples that come to mind are appreciated.

• Ed, can you be more explicit about what to you want? References to published work? Points of view? The problem was exploited in the nineties as a mine of examples of "supersymmetry" in quantum mechanics, as well as "scale invariance".
– arivero
Aug 19 '15 at 15:28
• Thanks @arivero. I am interested in references in which a potential V(x)=A delta(x) with A a constant is first used inside a non-relativistic Schrödinger Equation. It could be a textbook or an article (or a set of either) where the delta barrier is first presented. I know that there is a text by S. Flügge that treats the delta potential in 1971. This is the oldest I've found, but it seems recent. I don't have access to this book so wonder what else could be out there. Does that clarify better? Thanks Aug 19 '15 at 15:49
• Ed, I see. Indeed the plain delta, and its cousin the dirac comb, is traditional textbook material. If you are about this, and not modern uses, I reluctantly agree that an historian has better opportunity to pinpoint the origin of the problem.
– arivero
Aug 19 '15 at 15:57
• note that the $\delta$ potential can appear also as solid state physics (the comb) and as mathematics of operators (theory of self-adjoint extensions).
– arivero
Aug 19 '15 at 15:59
• @arivero. Thanks. I guess the super-users have spoken infinite wisdom and feel this should be in this history exchange. In any case, tracking further down, I found a reference from I. R. Lapidus in American Journal of Physics 37, 930 (1969) explicitly solving the problem, and referring to it as standard. So once again, there is going to be something older. Aug 20 '15 at 9:04

After looking for a while this is what I have found, although it does not mean that perhaps earlier references do not exist.

In the review article by M. Belloni and R. W. Robinett,

"The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics", PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS Volume: 540 Issue: 2 Pages: 25-122 Published: JUL 10 2014,

there is a section in which the delta potential is discussed at some length. They seem to find the first references to the use of the delta barrier in the classic article by R. de L. Kronig and W. G. Penney

"Quantum Mechanics of Electrons in Crystal Lattices" Proc. R. Soc. Lond. Ser. A Volume: 130 Pages: 499-513 Published: 1931.

There, the delta potential is introduced as the limit of zero width but constant area of square potential barriers.

Morse and Feshbach's treatise "Methods of Theoretical Physics" in v. 2 (published in 1953) discusses the single delta potential problem (as a well). Belloni and Robinett above make a note of this.

Intriguingly, I have never used references prior to the eighties. The oldest I used were:

R.G. Newton Inverse scattering by a local impurity in a periodic potential in one dimension, JMP 24 (1983)

P. Seba, The generalized point interaction in one dimension, Cz J Phys B 36:667 (1986)

M. Carreau; Four-parameter point-interaction in 1D-quantum systems, J Phys A: Math Gen, v 26 (1993)

There was a book on the topic in 1988, by Albeverio et al, that instead closing the topic, caused some subsequent work due to issues on the denomination of the different solutions. While the plain "delta barrier" was clear to everyone, some different barriers were expected to appear when using the $\delta '$ distribution, and it was controversial if the book had choosen the right solution.

Mathematically the question is about self-adjoint extensions of hermitian operators, and I'd expect it to appear in textbooks addressing this issue.