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European Journal of Physics

The infinite square well potential and the evolution operator method for the purpose of overcoming misconceptions in quantum mechanics

L. M. Arévalo Aguilar, Fernando Velasco Luna, Carlos Ignacio Robledo-Sánchez, M. L. Arroyo Carrasco

Published Jan 17, 2014• 16 citations
Publisher

Abstract

This paper explains and illustrates the application of the evolution operator method to solve problems in quantum mechanics. Currently, this method has been proposed as a useful way to overcome some misconceptions in quantum mechanics. To illustrate the method, we apply it to analyze and study the case of a quantum system inside an infinite square well potential (ISWP), and compare this result with that obtained using the traditional method. Also, we analyze the collapse and revival phenomenon in the ISWP. In this case, we argue that the usual approach to studying this effect requires one to extend the function's domain to infinity; however, there has not been any assurance that this extension preserves the self-adjointness of the Hamiltonian operator. The self-adjointness of the Hamiltonian operator is a vital requirement to guarantee the uniqueness of the Schrödinger equation's solution.

DOI
10.1088/0143-0807/35/2/025001

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This paper explains and illustrates the application of the evolution operator method to solve problems in quantum mechanics. Currently, this method has been proposed as a useful way to overcome some misconceptions in quantum mechanics. To illustrate the method, we apply it to analyze and study the case of a quantum system inside an infinite square well potential (ISWP), and compare this result with that obtained using the traditional method. Also, we analyze the collapse and revival phenomenon in the ISWP. In this case, we argue that the usual approach to studying this effect requires one to extend the function's domain to infinity; however, there has not been any assurance that this extension preserves the self-adjointness of the Hamiltonian operator. The self-adjointness of the Hamiltonian operator is a vital requirement to guarantee the uniqueness of the Schrödinger equation's solution.
Recently, there have been many efforts to use the research techniques developed in the field of physics education research to improve the teaching and learning of quantum mechanics. In particular, part of this research is focusing on misconceptions held by students. For instance, a set of misconceptions is associated with the concept of stationary states. In this paper, we argue that a possible way to remove these is to solve the Schrödinger equation using the evolution operator method (EOM), and stress the fact that to find stationary states is only the first step in solving that equation. The EOM consists in solving the Schrödinger equation by direct integration, i.e. Ψ( x , t ) = U ( t )Ψ( x , 0), where is the time evolution operator, and Ψ( x , 0) is the initial state. We apply the evolution operator method in the case of the harmonic oscillator.
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
Abstract We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity.
ABSTRACT: The main aim of this article is to present the time-evolution as the unitary and self-adjoint operators on Hilbert spaces and to describe its application in the development of quantum mechanics. The initial value problems associated with the quantum mechanical Schrodinger equation, ( ) = ( ) in the Hilbert space is solved by the use of timeevolution. The importance of time-evolution is also seen as the operation of turning machine in quantum mechanics, which is regarded as the time-evolution of the machine control state. Time-evolution of a quantum mechanical system is observed to be unitary and self-adjoint operators of Hilbert space, since it corresponds to an observable, specifically energy position and momentum. It was observed that time-evolution of a quantum mechanical system is generated by a self-adjoint operator, called Hamiltonian, V K H ˆ ˆ ˆ   , expressed by the Schrodinger equation, above.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.