The Dirac Delta Function

Marcos Amaku, Francisco Antônio Bezerra Coutinho, O. J. P. Éboli +1 more

R. A. Brewster, J. D. Franson

The Dirac delta function δ(x) is widely used in many areas of physics and mathematics. Here we consider the generalization of a Dirac delta function to allow the use of complex arguments. We show that the properties of a generalized delta function are very different from those of a Dirac delta function and that they behave more like a pole in the complex plane. We use the generalized delta function to derive the Glauber-Sudarshan P-function, P(α), for a Schrödinger cat state in a surprisingly simple form. Aside from their potential applications in classical electromagnetism and quantum optics, these results provide insight into the ability of the diagonal P-function to describe density operators with off-diagonal elements.

R.E. Moss

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, F. Masafumi Toyama

There is a situation such that, when a function ƒ( ) is combined with the Dirac delta function δ( ), the usual formula does not hold. A similar situation may also be encountered with the derivative of the delta function δ'( ), regarding the validity of . We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.