Instantaneous Reflection and Transmission Coefficients and a Special Method to Solve Wave Equation

TL;DR

This paper introduces a method to calculate instantaneous reflection and transmission coefficients for wave equations with varying potentials, using an adapted WKB approach called IWKB, enabling analytical solutions for complex potentials.

Contribution

The paper presents the Instantaneous WKB (IWKB) method, a novel approach to evaluate wave coefficients point-by-point in varying potentials, extending traditional WKB techniques.

Findings

Derived analytical expressions for instantaneous reflection and transmission coefficients.

Demonstrated the applicability of IWKB to complex, well-behaved potentials.

Provided a framework for solving wave equations analytically with variable potentials.

Abstract

People are familiar with quantum mechanical reflection and transmission coefficient. In all those cases corresponding potentials are usually assumed as of constant height and depth. For the cases of varying potential, corresponding reflection and transmission coefficients can be found out using WKB approximation method. But due to change of barrier height, reflection and transmission coefficients should be changed from point to point. Here we show the analytical expressions of the instantaneous reflection and transmission coefficients. Here as if we apply the WKB approximation at each point, so we call it as Instantaneous WKB method or IWKB method. Once we know the forward and backward wave amplitudes we can find out corresponding wave function by calculating Eiconal. For the case of analytically complicated potential corresponding differential equation seems to be unsolvable analytically. If the potentials are well behaved then obviously these could be replaced by functions of simple expression of same behaviour which can be integrated analytically and the equation is now possible to solve analytically.