An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein’s addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.

• Received 17 May 1994

DOI:https://doi.org/10.1103/PhysRevA.50.3627