Comment on “On the energy levels of a finite square-well potential” [J. Math. Phys. <b>41</b>, 4551 (2000)]

Aronstein, David L.; Stroud, C. R.

doi:10.1063/1.1322081

Cited by 8 publications

(6 citation statements)

References 9 publications

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“…Naturally, there are no explicit expressions for the SCGF and the rate function, since the former is obtained from a transcendental equation. However, we can easily derive asymptotics for both functions using known asymptotics for the energy levels in the infinite depth limit [54][55][56][57]. For the SCGF, we find…”

Section: Combined Solutionmentioning

confidence: 98%

Dynamical phase transition in drifted Brownian motion

Nyawo

Touchette

2018

Phys. Rev. E

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We study the occupation fluctuations of drifted Brownian motion in a closed interval, and show that they undergo a dynamical phase transition in the long-time limit without an additional low-noise limit. This phase transition is similar to wetting and depinning transitions, and arises here as a switching between paths of the random motion leading to different occupations. For low occupations, the motion essentially stays in the interval for some fraction of time before escaping, while for high occupations the motion is confined in an ergodic way in the interval. This is confirmed by studying a confined version of the model, which points to a further link between the dynamical phase transition and quantum phase transitions. Other variations of the model, including the geometric Brownian motion used in finance, are considered to discuss the role of recurrent and transient motion in dynamical phase transitions.

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Section: Combined Solutionmentioning

confidence: 98%

Dynamical phase transition in drifted Brownian motion

Nyawo

Touchette

2018

Phys. Rev. E

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“…We conclude this section by pointing out that explicit analytical expressions for the energies of the finite square well are also available [20][21][22][23], but will not be considered here, because their derivation and further manipulation requires techniques not found in the mathematical repertoire of a physics undergraduate; we will, however, quote and use an approximation that is valid for the case of small γ (see below).…”

Section: A Retrospectmentioning

confidence: 99%

“…Paul and Nkemzi [21] showed that the explicit expression for E n can be expanded as a series in powers of γ; when their result is corrected for an algebraic error, identified by Aronstein and Stroud [22], the expression for the energy comes out to be…”

Section: Finite and Infinite Wellsmentioning

confidence: 99%

The finite square well: whatever is worth teaching at all is worth teaching well

Naqvi,

Waldenstrøm

2015

Preprint

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A contemporary physicist would be hard put to agree entirely with the author of a 1959 textbook on quantum mechanics, who wrote: "A second simple, one-dimensional system, somewhat divorced from reality but illustrative of the principles of the theory, is a particle in a box with finite walls." Interest in this prototypic system has not diminished over the years, and is not likely to do so in the near future, because it has now been wedded to reality and is seen as a paradigm for quantum wells and other nanosystems. The treatment of this topic in standard textbooks has changed little over the last few decades, and such changes as have come to our notice leave, in our opinion, much to be desired. The purpose of this article is to enable, even encourage, other teachers to give the finite square well the attention that it deserves; to this end, the authors provide a tutorial review that is more instructive and comprehensive than the accounts presented in numerous textbooks and dispensed in still more numerous online resources, but does not make any additional demands on the mathematical abilities of the student. Given the elemental nature of the topic, the authors claim no originality; with nothing more than the winnowing fork in their hands, they can only clear the threshing floor, gather the wheat into the barn, and let others decide whether or no the chaff should be burnt with unquenchable fire.

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“…The wave function satisfies the equation (8) in region 1, and the equation (9) in region 3, for some complex constants C and D, to be determined.…”

Section: Solution Using Lambert Wmentioning

confidence: 99%

“…After deriving a pair of equations to describe the bound energy levels within the well, the solution is carried out by graphical or computational methods. It is sometimes said that the FSW problem does not have an exact solution, but there are in fact exact solutions as presented in the papers of Burniston and Siewert [4][5][6] and others [7][8][9][10][11][12]. Those exact methods generally rely upon contour integration in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

Tutorial: The quantum finite square well and the Lambert W function

Roberts

Valluri

2017

Can. J. Phys.

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We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping w → z = we w between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. There are many opportunities to follow up, and we present the method in a pedagogical manner to stimulate further research in this and related avenues.

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Comment on “On the energy levels of a finite square-well potential” [J. Math. Phys. 41, 4551 (2000)]

Cited by 8 publications

References 9 publications

Dynamical phase transition in drifted Brownian motion

Dynamical phase transition in drifted Brownian motion

The finite square well: whatever is worth teaching at all is worth teaching well

Tutorial: The quantum finite square well and the Lambert W function

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