Approximate Approaches to the One-Dimensional Finite Potential Well

Autor/inn/en	Singh, Shilpi; Pathak, Praveen; Singh, Vijay A.
Titel	Approximate Approaches to the One-Dimensional Finite Potential Well
Quelle	In: European Journal of Physics, 32 (2011) 6, S.1701-1710 (10 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0143-0807
DOI	10.1088/0143-0807/32/6/023
Schlagwörter	Quantum Mechanics; Evaluation; Physics; Comparative Analysis; Problem Solving; Teaching Methods; College Students; College Science; Equations (Mathematics); Science Education; Foreign Countries; India + Suchen Sie Ihr Suchwort? Quantenmechanik; Evaluierung; Physik; Problemlösen; Teaching method; Lehrmethode; Unterrichtsmethode; Collegestudent; Equations; Mathematics; Gleichungslehre; Naturwissenschaftliche Bildung; Ausland; Indien
Abstract	The one-dimensional finite well is a textbook problem. We propose approximate approaches to obtain the energy levels of the well. The finite well is also encountered in semiconductor heterostructures where the carrier mass inside the well (m[subscript i]) is taken to be distinct from mass outside (m[subscript o]). A relevant parameter is the mass discontinuity ratio beta = m[subscript i]/m[subscript o]. To correctly account for the mass discontinuity, we apply the BenDaniel-Duke boundary condition. We obtain approximate solutions for two cases: when the well is shallow and when the well is deep. We compare the approximate results with the exact results and find that higher-order approximations are quite robust. For the shallow case, the approximate solution can be expressed in terms of a dimensionless parameter sigma[subscript l] = 2m[subscript o]V[subscript 0]L[superscript 2]/h-bar [superscript 2] (or sigma = beta[superscript 2]sigma[subscript l] for the deep case). We show that the lowest-order results are related by a duality transform. We also discuss how the energy upscales with "L" (E is proportional to 1/L[superscript gamma]) and obtain the exponent gamma. Exponent gamma [right arrow] 2 when the well is sufficiently deep and beta [right arrow] 1. The ratio of the masses dictates the physics. Our presentation is pedagogical and should be useful to students on a first course on elementary quantum mechanics or low-dimensional semiconductors. (Contains 2 tables, 3 footnotes and 4 figures.) (As Provided).
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Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10