On Quantum-Mechanical Reflection Coefficients and their Numerical Determination

Abstract

A method, based on the use of Milne's function $w$, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point $x$ in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.