# Wentzel–Kramers–Brillouin quantization rules for two-dimensional quantum dots

TL;DR: In this paper, the Wentzel-Kramers-Brillouin (WKB) approximation is applied to an electron in a central force potential, confined in a two-dimensional disc, and the quantization rules for such a system are obtained.

Abstract: In this paper, we apply the semiclassical Wentzel–Kramers–Brillouin (WKB) approximation to an electron in a central force potential, confined in a two-dimensional disc, and we obtain the quantization rules for such a system. As explicit examples, we consider the two most widely studied potentials, viz., the parabolic potential (the harmonic oscillator) and the Hydrogenic impurity state (Coulomb potential) in two dimensions. Both the systems are confined within an impenetrable circular wall of radius r0: In particular, we determine the energies as well as eigenfunctions by this approach. On comparing these energies with those from exact numerical values, the agreement is found to be quite good. Moreover, the WKB approach is found to give a good estimate of the wave functions as well. These results suggest that the WKB approximation works well even for such rigid wall spatial confinement and the present approach can be applied to other confined systems such as those which are encountered in mesoscopic physics. r 2002 Elsevier Science B.V. All rights reserved.

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TL;DR: In this paper, the ground state binding energy of a hydrogenic impurity located at the center of a quantum dot has been studied with a variational approach, and it has been found that a variation in the binding energy has depended on the geometry of the dot.

Abstract: The binding energy of a hydrogenic impurity of a multilayered spherical GaAs-(Ga,Al)As quantum dot has been investigated as a function of the barrier thickness and the inner dot thickness for various barrier potentials in the effect of the band non-parabolicity. Within the effective mass approximation, the ground state energy has been calculated using the fourth-order Runge–Kutta method. The ground state binding energy of hydrogenic impurity located at the center of a quantum dot has been studied with a variational approach. We have found that a variation in the binding energy has depended on the geometry of the dot.

58 citations

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TL;DR: In this article, the energy spectrum and polarizabilities of a hydrogen atom confined to a sphere of radius R are analyzed in terms of the numerical approach, model wave functions, and simple analytical expressions, which provide a useful description of these properties.

Abstract: The energy spectrum and polarizabilities of hydrogen atom confined to a sphere of radius R , are analysed in terms of the numerical approach, model wave functions, and simple analytical expressions, which provide a useful description of these properties. The scaling relations are used to develop simple expressions for the energies of the confined helium atom in terms of screening effect. The considerations are extended to the hydrogen atom in an oscillator potential, and to off-centre confinement. The general results provide a clear understanding of the implications of confinement.

31 citations

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TL;DR: In this article, a formalism was developed to obtain the energy levels of the electron in a central force potential confined in a spherical quantum dot with radius rC by the proper quantization rule and the Wentzel-Kramers-Brillouin approximation.

Abstract: In this article, we develop a formalism to obtain the energy levels of the electron in a central force potential confined in a spherical quantum dot with radius rC by the proper quantization rule and the Wentzel-Kramers-Brillouin approximation. It is shown that the numerical results are in good agreement with exact solutions. To illustrate this method, we consider the linear harmonic oscillator and Coulomb potential confined within an impenetrable sphere of radius rC in three dimensions. © 2013 Wiley Periodicals, Inc.

15 citations

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TL;DR: In this article, the analytical transfer matrix method (ATMM) was applied to calculate energy eigenvalues of a particle in low dimensional sharp confining potential for the first time, and deduced the quantization rules of this system.

Abstract: This paper applies the analytical transfer matrix method (ATMM) to calculate energy eigenvalues of a particle in low dimensional sharp confining potential for the first time, and deduces the quantization rules of this system. It presents three cases in which the applied method works very well. In the first quantum dot, the energy eigenvalues and eigenfunction are obtained, and compared with those acquired from the exact numerical analysis and the WKB (Wentzel, Kramers and Brillouin) method; in the second or the third case, we get the energy eigenvalues by the ATMM, and compare them with the EBK (Einstein, Brillouin and Keller) results or the wavefunction outcomes. From the comparisons, we find that the semiclassical method (WKB, EBK or wavefunction) is inexact in such systems.

11 citations

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TL;DR: In this article, an alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius r 0, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation.

Abstract: An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius r
0, in the framework of Wentzel–Kramers–Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert: the centrifugal term is expanded perturbatively (in powers of ℏ), decomposing it into 2 terms – the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energies of the hydrogen atom, to the first order in ℏ, with all higher order corrections vanishing. In the present study, this formalism is extended to the case of radial potentials under hard wall confinement, to check whether the same argument holds good for such confined systems as well. As explicit examples, 3 widely known potentials are studied, which are of considerable importance in the theoretical treatment of various atomic phenomena involving atomic transitions, namely, the 3-dimensional harmonic oscillator, the hydrogen atom and the Hulthen potential.

9 citations

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01 Jan 1943TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform

Abstract: 0 Introduction 1 Elementary Functions 2 Indefinite Integrals of Elementary Functions 3 Definite Integrals of Elementary Functions 4.Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integrals of Special Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequalities 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform

27,354 citations

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6,036 citations

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TL;DR: In the fourth edition of the book as mentioned in this paper, the section on definite integrals has been considerably enlarged and a section added on integrals essentially of elliptic type, which is the case with the present edition.

Abstract: By Herbert Bristol Dwight New York. London: The Macmillan Company. Pp. x + 336. Price 26s. A deservedly popular work of reference, first published almost thirty years ago, this book now appears in a fourth edition, re-designed and reset. The section on definite integrals has been considerably enlarged and a section added on integrals essentially of elliptic type.

2,404 citations