Semiclassical calculation of oscillator-strengths

A semi-empirical model for computing radial matrix elements in one-electron atoms and ions

“…In this way, van Regemorter et al [15] obtain an easily programmable expression for (vl I r ln' l' ), which after a linear interpolation provides any Coulomb matrix elements. Alternatively, matrix elements may be evaluated within the JWKB approximation using saddle-point (9) 2.084 (9) 2.082 (9) integration [22], with good accuracy even for large energy variation. Besides, a JWKB representation of the wavefunctions leads to Coulomb matrix elements in a closed form, involving Anger functions and suitable average values v, and l~.…”

Semiclassical matrix mechanics. I. The harmonic oscillator

“…In this paper we discuss only the harmonic oscillator and in this case it is the extraordinary accuracy of the resulting matrices which gives evidence for our computational scheme. However the modified semiclassical rules also succeed for many other systems [1][2][3].…”

“…In this paper we have included a factor 1/2 in the wave-function and for this reason equations (17), (18) do not need the factor 1/4 used in references [1][2][3][4]. The normalization Nn of equation (8) has the consequence that Unn = 1 for all n. Equations (17) and (18) are semiclassical in the sense that the WKB functions IF n (, ) are explicitly constructed from the classical velocity and action functions without ever solving a Schroedinger equation.…”

“…We therefore take the point of view that equations (17), (18) constitute basic assumptions of a new semiclassical theory, and we will examine the consequences of these assumptions. The physical interpretation of equations (17), (18) is developed in greater detail in references [1][2][3][4]. 4.…”

“…To this end we have explored the calculation of quantities which arise in the Heisenberg and Feynman formulations of quantum mechanics using the WKB approximation to the Schroedinger wave-functions. In this paper we consider a simple specific system, the onedimensional harmonic oscillator, and form the semiclassical matrix-elements following a new calculational scheme based on a contour-integral inner product of WKB wave-functions [1][2][3][4]. The results are surprising.…”