Resonant-State Expansion Applied to Non-Relativistic Wave Equation in One-Dimension
DOI:
https://doi.org/10.56919/usci.2434.021Keywords:
Resonant-states expansion, unperturbed basis, non-relativistic wave equation, perturbation theoryAbstract
Study’s Excerpt
- The rigorous resonant-state expansion (RSE) method is extended to the non-relativistic one-dimensional wave equation.
- Resonant states (RSs) wave numbers as the unperturbed basis were employed to confirm the RSE's convergence to exact solutions.
- RSE's has potentials for systematically calculating RSs in complex quantum systems with multi-well potentials.
Full Abstract
The resonant-state expansion (RSE), a rigorous perturbation theory recently developed in electrodynamics, is here applied to the non-relativistic wave equation in one-dimension. The resonant states (RSs) wave numbers for the double well system are analytically calculated and used as the unperturbed basis for calculating the RSE. We demonstrate the efficiency of the RSE by verifying its convergence to the exact solution for a triple well potential. We show that for the chosen perturbations (i.e., for and ), the method is particularly suitable for calculating all the RSs within the spectrum.
References
Armitage, L. J.,Doost, M. B., Langbein, W., and Muljarov, E. A. (2014). Resonant state expansion applied to planer waveguides. Phys. Rev. A 89, 053832. https://doi.org/10.1103/PhysRevA.89.053832
Doost, M. B., Langbein, W., and Muljarov, E. A. (2012). Resonant state expansion applied to planer open optical systems.Phys. Rev. A 85, 023835. https://doi.org/10.1103/PhysRevA.85.023835
Gamow, G. (1928). Zurquantentheorie des atomkernes. Zeitschriftfur physic. 51, pg. 204-212. https://doi.org/10.1007/BF01343196
Hatano, N. (2008). Some properties of the Resonant states in quantum mechanics and its computations. Prog. Theor.Phys. 119, pg. 187. https://doi.org/10.1143/PTP.119.187
Lind, P. (1992). Completeness relations and resonant state expansion. Physical Review C 47, pg. 1903. https://doi.org/10.1103/PhysRevC.47.1903
Mostert, M. (2014). Resonant states and Resonant states expansion in Quantum Mechanics, Year 4 Physics report
Muljarov, E. A., Langbein, W., and Zimmermann, R. (2010). Brillouin-Wigner Perturbation theory in open electromagnetic systems.EPL 92, pg. 5. https://doi.org/10.1209/0295-5075/92/50010
Siegert,A. J. F. (1939). On the Derivation of the Dispersion Formula for Nuclear Reactions. Physical Review 56, pg. 750. https://doi.org/10.1103/PhysRev.56.750
Tanimu, A., and Muljarov, E. A. (2018). Resonant state expansion applied to one-dimensional quantum systems. Phys. Rev. A 98, 022127. https://doi.org/10.1103/PhysRevA.98.022127
Tanimu, A., and Muljarov, E. A. (2018). Resonant states in double and triple quantum wells. J. Phys. Commun. 2, 115008. https://doi.org/10.1088/2399-6528/aae86a
Tanimu, A., and Bagudo, I. M. (2020). Resonant states in a one-dimensional quantum system.FJS.4, pg. 300. https://doi.org/10.33003/fjs-2020-0403-394
Tanimu, A., and Bagudo, I. M. (2022). Resonant states expansion of the one-dimensional Schrödinger equation. BAJOPAS. 13, pg. 320.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Abdullahi Tanimu, Ibrahim Muhammad Bagudo, Haruna Abdullahi Abubakar
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
