Numerical investigations of symmetric embedded quantum well systems using the shooting method and transfer matrix technique


Embedded quantum well systems are constructed via a superposition of modified Pöschl‐Teller (MPT) potentials. To illustrate the advantages and/or disadvantages of each system, a comparison is made on their energy spectra and scattering properties. The bound states of each system are obtained using the numerical shooting method. The accuracy of this method depends on the step length δz, which can be completely controlled, allowing an improved comparison with analytical solutions to be achieved. A systematic approach of finding the eigenenergies En (n = 0, 1, 2, 3, …) is also developed in this work. The approach utilizes the established bisection method—a root‐finding algorithm that repeatedly divides an interval in half then selects the subinterval in which a root exists. Here selective bisection is implemented for computational efficiency. This entire procedure in finding the solutions of the Schrödinger equation is tested for a single MPT well where the solution is analytically known. The result shows a perfect agreement between the numerical and the analytical method. The transfer matrix approach is used to obtain the transmission probability of a particle passing through the embedded quantum well systems. This technique also shows a perfect agreement with the analytical solution.