This paper continues the work done in previous works: [1] and [2]. We demonstrate an algebraic method for finding the exact time evolution operator of a system with two electric fields, one that is constant and another one that is oscillating, under the velocity gauge for the time dependent Schrödinger equation. We start off by producing the Schrödinger equation under the said velocity gauge, as well as establishing the algebraic method to be used to obtain the exact time evolution operator. Subsequently, the Schrödinger equation is developed for our system under the established method. The algebraic method used in this paper was developed by Wei and Norman [7] and was employed in this paper to find the Gordon-Volkov states linearly. This represents an advantage since, classically, these states are found using ansatz methods, as is demonstrated in the article written by Lefebvre [3], which was the inspiration for the parameters used in the system presented here.

time evolution operator, algebraic approach, Gordon-Volkov states, velocity gauge.

Received: January 2, 2023; Revised: January 30, 2023; Accepted: February 1, 2023; Published: March 20, 2023

How to cite this article: Alejandro Palma and Isaias Lopez-Garcia, Time dependent Schrödinger equation: III. Velocity gauge, Far East Journal of Applied Mathematics 116(1) (2023), 55-60. http://dx.doi.org/10.17654/0972096023004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] Alejandro Palma, Time-dependent Schrödinger equation: I. Longitude gauge, Far East J. Appl. Math. 113 (2022), 29-35.
[2] Alejandro Palma, Time-dependent Schrödinger equation: II. Reduced velocity gauge, Far East J. Appl. Math. 113 (2022), 37-43.
[3] R. Lefebvre, Resonant tunneling in the presence of two electric fields: One static and the other oscillating, International Journal of Quantum Chemistry 80 (2000), 110-116.
[4] M. Büttiker and R. Landauer, Traversal time for tunneling, Phys. Rev. Lett. 49(23) (1982), 1739-1742.
[5] M. Lazarevic, Stability and stabilization of fractional order time delay systems, Scientific Technical Review 61 (2011), 31-45.
[6] R. K. Mains and G. I. Haddad, Time-dependent modeling of resonant-tunneling diodes from direct solution of the Schrödinger equation, J. Appl. Phys. 64 (1988), 3564.
[7] James Wei and Edward Norman, Lie algebraic solution of linear differential equations, J. Math. Phys. 4 (1963), 575-581.
[8] P. Zhang and Y. Lau, Ultrafast strong-field photoelectron emission from biased metal surfaces: exact solution to time-dependent Schrödinger equation, Sci. Rep. 6 (2016), 19894.