A physically important class of integrals expressed as a parameter derivative |
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Authors: | Per Olof Frö man |
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Affiliation: | Department of Theoretical Physics, University of Uppsala, Box 803, S-751 08 Uppsala, Sweden |
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Abstract: | 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-dimensional Schrödinger 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. |
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Keywords: | 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 |
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