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arXiv:1403.6685 · Canadian Journal of Physics

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

Ken Roberts, S.R. Valluri

Published Nov 17, 2016• 30 citations
arXivHTMLPDF

Abstract

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 = wew 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. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.

DOI
10.1139/cjp-2016-0602

Related papers

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

Valluri, Sree Ram, Roberts, Ken

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. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.

•Feb 1, 2017•31 citations

Lambert W Lines and Finite Square Well Sensors

Roberts, Ken, N. Jisrawi, J. Jeyasitharam +3 more

The bound state energies of a 1-dimensional finite quantum square well (FSW) can be determined using a geometric method, involving a smooth mapping between two copies of the complex plane. The method allows one to identify particular strengths of the FSW at which the system can become unusually sensitive to changes in the well depth or geometry. In the present paper we explore that sensitivity, and exhibit a 3-D visualization of the solutions.

arXiv:2210.07359•Oct 12, 2022•0 citations

Solving Schrödinger Equation for Finite Potential Well Using the Iterative Method

Laith Abdul Aziz Al-Ani, Russul K. Abid

The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.

Journal of Al-Nahrain University-Science•Dec 1, 2019•2 citations

On a solution method for the bound energy states of a particle in a one-dimensional symmetric finite square well potential

Orion Ciftja, Brett Johnston

Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.

European Journal of Physics•Apr 17, 2019•6 citations

A simpler graphical solution and an approximate formula for energy eigenvalues in finite square quantum wells

F. M. S. Lima

The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.

American Journal of Physics•Oct 21, 2020•12 citations

An introduction to the inverse quantum bound-state problem in one dimension

Thomas D. Gutierrez

A technique to reconstruct one-dimensional, reflectionless potentials and the associated quantum wave functions starting from a finite number of known energy spectra is discussed. The method is demonstrated using spectra that scale like the lowest energy states of standard problems encountered in the undergraduate curriculum, such as the infinite square well and the simple harmonic oscillator.

arXiv:1308.4742•American Journal of Physics•Jun 24, 2014•3 citations
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 = wew 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. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.
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. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector.
The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books.A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well.Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions.In this work a computational techniques is adopted to find the exact solution for FSW.To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM).Six values of potential width have been studied with potential depth.The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states.Six values of potential depth have been studied with potential width.The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states.A Comparison Between Finite and Infinite Potential Well has been also presented.The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well.In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.
Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.
The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.