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Molecular Physics

Exact analytical solution of time-independent Schrödinger equation for a system consisting of two flat potentials coupled by a rectangular potential

Proma Mondal, Aniruddha Chakraborty

Published Aug 28, 2021• 0 citations
Publisher

Abstract

We have proposed an analytical method for finding the exact solution for the two-state scattering problems, where two flat potentials are coupled by a rectangular potential. We found a way to express the two-state problem by an effective single-state problem, where it is easy to estimate the transition probability from one potential to the other by estimating the amount of missing flux from the first potential. This has been an unsolved problem for a long time and is of considerable importance as a model for non-adiabatic electronic transition.

DOI
10.1080/00268976.2021.1968055

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We have proposed an analytical method for finding the exact solution for the two-state scattering problems, where two flat potentials are coupled by a rectangular potential. We found a way to express the two-state problem by an effective single-state problem, where it is easy to estimate the transition probability from one potential to the other by estimating the amount of missing flux from the first potential. This has been an unsolved problem for a long time and is of considerable importance as a model for non-adiabatic electronic transition.
Non-adiabatic transition probability with a moving δδδ potential coupling
The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving δδδ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving δδδ potential and an expression for non-adiabatic transition probability has been derived.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.