Numerical transforms from position to momentum space via Gaussian quadrature in the complex plane

Abstract

A method for the numerical evaluation of integrals involving Bessel functions of the first kind, via non-standard Gaussian quadrature, is appraised. This method uses specialized Gaussian quadratures based on weight functions which are modified Bessel functions of the third kind. We apply the method to calculate momentum space radial wave functions of the two-dimensional hydrogen atom and harmonic oscillator, via two-dimensional Fourier or Hankel transforms of the position space functions. Results show that the method performs best for the asymptotic or large p regions of the hydrogen atom. We also illustrate that accurate results can be obtained for the smaller argument regions with the use of higher-order quadratures. On the other hand, the method fails in its application to the harmonic oscillator. A discussion of the strengths and weaknesses of the method is presented.