On the Evaluation of Certain Integrals Important in the Theory of Quanta

Abstract

It is known [1] that the matrix of the hydrogen atom, determining the intensities of the hydrogen series lines and of their fine structure components, essentially depends on the integral [equation (1)], where the functions χ and χ' are solutions of the Schroedinger equation [2] [equation (2)]. The symbols µ, e stand for the mass and the charge of the electron; K = h/2π (h Planck's constant). The energy E and the integer k have different values in χ and χ'. These functions are supposed to be finite for r = 0 and to vanish for r = ∞. Since in the case of elliptic orbits the functions X, X' turn out to be polynomials multiplied by an exponential, the direct evaluation term by term is possible on principle. The numerical computations involved are, however, so lengthy as to make this method almost prohibitive in practice. We give, therefore, in this note a reduction of the integral (1) to a simple and convenient expression. Such a reduction is quite indispensable in the case of hyperbolic orbits. Moreover, the procedure applied has an interest beyond the special case of the Kepler motions, since quite analogous expressions occur in other problems of the quantum theory. In fact, the same method has been used by the author for reducing the intensity expressions of the components in the Stark effect.3 Only the simple closed Kepler motion, neglecting relativity effect, and spin of the electron, will be considered in the following sections.

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