The theory of transmission and reflection is widely used in Physics and Engineering. Although many approaches have been published on the subject, it seems possible to present new methods to study it.
Using the well-known matrix formulation of the reflection and transmission of electromagnetic waves by a stratified planner structure, we show that the reflection and transmission coefficients of any number of isotropic media can be written without any calculation by using a simple general formula. This formula uses the so-called elementary symmetric functions that are extensively used in the mathematical theory of polynomials. Although the results are presented with the use of the language of electromagnetic theory, they are quite general and can be applied to other fields of physics that use the theory of reflection and transmission. They can be applied to matter waves satisfying Schrodinger equation. We show that the reflection and transmission coefficients of any number of quantum wells or barriers can be written in the same way. As a test of the approach, we apply the results to a celebrated problem in quantum mechanics, rectangular barrier. Finally, one-dimensional scattering from a series of delta-function barriers (Dirac Comb system) is considered in the context of nonrelativistic quantum mechanics. A simple closed-form expression for the transmission probability obtained recently by David J. Griffiths and Nicholas F. Taussig is presented. This approach uses the transfer matrix and Chebychev polynomials of the second kind. Next we present a new approach for that system using the elementary symmetric functions. This approach is easier to handle than the previous one and it treats the quantum scattering from the electromagnetic theory point of view. The results of the new method are compared with those of the previous one.