 
Summary: Numerical integration of functions originating from
quantum mechanics
R. Armiento
Department of Physics, Royal Institute of Technology, AlbaNova University Centre,
SE106 91 Stockholm, Sweden
Applications in quantum physics commonly involve large batches of integrals of smooth but very
oscillatory functions. The purpose of this work is to benchmark and compare different numerical
algorithms for evaluating such integrals. The routines studied include: two from the QUADPACK
package based on GaussKronrod quadrature; one routine based on Patterson's improvements
of GaussKronrod quadrature; and two routines that use a nonstandard algorithm of applying
quadraturelike rules of unrestricted order. The last algorithm has been seen in previous works,
but is not in widespread use. The present work includes optimized implementations of this
algorithm for both serial and parallel computation.
1. BACKGROUND
Applications performing quantum physics based calculations usually involve nu
merical treatment of `wave functions'. These functions originate from the wavelike
Sch¨odinger's differential equation and are smooth but very oscillatory. Although it
is possible for these functions to involve difficult or singular points, the locations
of such points relate closely to physical properties of the treated problem, and are
assumed to be known in advance. It is not uncommon for applications to calculate
