A physically important class of integrals expressed as a parameter derivative
2007 (English)In: Annals of Physics, ISSN 0003-4916, E-ISSN 1096-035X, Vol. 322, no 9, 2145-2167 p.Article in journal (Refereed) Published
The paper deals with a class of integrals, the integrands of which contain the square of a solution of a second-order linear, ordinary differential equation. Such integrals often arise in quantum mechanics as normalization integrals or expectation values. A generalized, unified procedure for rewriting such an integral, associated with a differential equation of the Sturm-Liouville type with unspecified boundary conditions, as a parameter derivative is presented. The formula thus obtained can be used for the evaluation of various integrals of physical interest. As an application we present a simplified derivation of a formula given by de Alfaro and Regge, in which the quantal normalization integral is expressed in terms of the Jost function. Other applications to integrals involving special functions and to integrals associated with the one-din-tensional Schrodinger equation are also presented. Furthermore, it is explained why an approximate formula for expectation values is much more accurate than one can expect from the usual, crude derivation of it, and why certain attempts to improve that derivation have failed.
Place, publisher, year, edition, pages
2007. Vol. 322, no 9, 2145-2167 p.
quantum mechanics, quantal normalization integrals, expectation values, differential equation of the sturm-liouville type, Bessel function, legendre function, Jost function, de Alfaro and regge normalization formula, Hellmann-Feynmann formula, scattering phase shift
IdentifiersURN: urn:nbn:se:uu:diva-143508DOI: 10.1016/j.aop.2006.09.005ISI: 000249550600009OAI: oai:DiVA.org:uu-143508DiVA: diva2:390305