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ERIC Number: EJ963914
Record Type: Journal
Publication Date: 2011-Nov
Pages: 10
Abstractor: As Provided
Reference Count: 14
ISSN: ISSN-0143-0807
Approximate Approaches to the One-Dimensional Finite Potential Well
Singh, Shilpi; Pathak, Praveen; Singh, Vijay A.
European Journal of Physics, v32 n6 p1701-1710 Nov 2011
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.)
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Publication Type: Journal Articles; Reports - Descriptive
Education Level: Higher Education
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A
Identifiers - Location: India