Phase-integral calculation of quantal matrix elements between unbound states, without the use of wavefunctions

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Abstract
The paper deals with the calculation of diagonal as well as off-diagonal matrix elements between unbound states pertaining to a radial Schrödinger equation. For the case that the effective potential, i.e. the potential with the centrifugal barrier included, is real and that only a single generalized classical turning point exists, we derive, on the basis of a phase-integral method, a new, general formula for the calculation of such matrix elements without the use of wavefunctions. The formula applies to an arbitrary order of the phase-integral approximations used. While conventional methods may present difficulties when the radial wavefunctions oscillate rapidly in space, this formula, in which the wavefunctions do not appear, is expected to be more accurate the more rapidly the wavefunctions oscillate. The formula is tested numerically for a Lennard-Jones potential and is found to yield extremely accurate results. When specialized to the first-order approximation, our formula goes over into a formula derived previously by other authors. They derived the formula by introducing the first-order JWKB approximation for the wave-functions in the integral defining the matrix element and simplifying the resulting expression by using simple qualitative arguments. The remarkably good accuracy of this first-order formula has puzzled previous authors. So has also the fact that attempts to improve it, by including quantities neglected on the basis of the simple qualitative arguments, and by using Airy functions or uniform approximations for the wavefunctions through the turning points, have failed. Our derivation throws light on the question of the accuracy to be expected, and it explains why already the first-order formula is remarkably accurate and why no improvement is gained in the attempts mentioned.