Quantum Graphs : Physics of Quantum Singularity (2-4)

Settings of Quantum Graphs 4

In the end, what we have, as the result of point interaction, is the mismatch between the wave functions at the right and left side of the singular point, and that mismatch can arise as discontinity of both wave function itself and its derivative.

So far so good, but where is the delta function we are familiar with? In the form we have dealt with up to now, we see no sign of it, or do you see any?

To make this point clear, we need to represent the mismatch in a different form using the "transfer matrix" \Lambda that connects the wave function of left and right side of the singularity as

(2.9)

A tedius but straightforward rewriting of (2.7) and (2.8) yields an explicit formula

(2.10)

Obviously, this form is meaningful only when the denominator \beta^* is nonzero. Therefore the formulae (2.6) and (2.8) are, in general, better mathematical representations of the whole set of possible point interaction.

Let us assume the case of \alpha_i = \beta_r = 0 in (2.10). If we further set \beta=\sin\xi, the unitarity condition allows two possible choices for the remaining parameter \alpha_r, namely, \alpha_r = \cos\xi, or \alpha_r = -\cos\xi. If we denote corresponding \Lambda as \Lambda_\delta and \Lambda_\epsilon respectively, they are given by

、

(2.11)

It is now easy to see that \Lambda_\delta is nothing but our usual "Dirac's delta function" potential that leaves the wavefunction continuous but cause its derivative discontinuous. The point interaction represented by \Lambda_\delta, on the other hand, leaves the derivative of wavefunction continuous but cause wave function itself discontinuous. This is what we call "epsilon function potential", which is also known by a misnomer "delta-prime" potential. They clearly form a very special small one-parameter subsets of entire quantum point interactions.

We sometimes encounter in literature, that even this "epsilon" or "delta-prime" potential is termed as nonstandard and treated as exotic object which is beyond experimentally realizable interaction. We shall detail in the following that this is simply wrong and entire point interaction beyond delta is of relevance to modern quantum mechanics.