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Research Article

Construction of a Rational Delta Function Using the Reverse Cantor Set and its Application to Quantum Mechanics via Pseudo-spectral Methods

Aadel A. Chaudhuri

*Trends in Applied Sciences Research,
2007, 2(1), 1-14.*

### Abstract

A simple yet novel method for construction of rational delta function using the reverse Cantor set and its application to quantum mechanics is presented. This study is primarily concerned with introduction or derivation of some simple rational delta function that represents the orthonormality condition of the computed eigenfunctions — both in the physical space as well as the Fourier space — which will serve as the basis functions for the wave function, Ψ, the solution to the Shrφdinger wave equation subjected to prescribed boundary conditions. The rational delta function, δ_{n }(x), based on a hitherto unavailable reverse Cantor set derived in this study, reduces to the Dirac δ-function and Kronecker δ-function in the limiting cases of n → ∞ and n → 0, respectively and thus bridges the gap between the two situations that arise in quantum mechanics, namely bound states with discrete eigenvalues and scattering case with continuous spectrum of eigenvalues. Most important, this novel rational delta function, δ_{n}(x), permits the resulting computed wave function to be expressed in the form of Discrete Fourier Transform (DFT) in the Fourier domain and recover the sampled wave function in the physical domain by employing the inverse discrete Fourier transform (IDFT). The example problem of a barrier inside a well studied here sheds new light on the nature of interaction of two or more potentials and will serve as a prelude to more complex many body interaction problems.

**ASCI-ID: 95-78**

Cited References Fulltext

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