Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Overcoming misconceptions in quantum mechanics with the time evolution operator

P. C. Garcia Quijas, L. M. Arévalo Aguilar

Published Jan 2, 2007• 27 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/28/2/001

Related papers

Factorizing the time evolution operator

P. C. García Quijas, L. M. Arévalo, Aguilar Centro +4 more

There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function, because it implies to apply terms like (A+B)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize. In this work we present useful ways to factorizing the time evolution operator when the argument of the exponential is a sum of two operators which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed in previous papers published in this Journal. Finally, we discuss the possible insight that can be learned using this approach in teaching Quantum Mechanics.

•Jan 1, 2006•15 citations

Comment on ‘Overcoming misconceptions in quantum mechanics with the time evolution operator’

F. Masafumi Toyama, Yukihisa Nogami

In their paper ‘Overcoming misconceptions in quantum mechanics with the time evolution operator’, García Quijas and Arévalo Aguilar (2007 Eur. J. Phys. 28 147) examined the time-dependent wave function of a particle in the one-dimensional harmonic oscillator potential using two different methods. The two wave functions that the authors obtained through the methods have different analytical expressions. The authors showed numerically that the two wave functions lead to the same probability density. When the real parts of the wave functions are compared, however, they are different in their details. That was puzzling because both wave functions are supposed to be solutions of the same time-dependent Schrödinger equation with the same initial condition. We point out that the two wave functions are actually identical. We show this analytically.

European Journal of Physics•Apr 22, 2013•7 citations

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 +1 more

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.

European Journal of Physics•Jan 17, 2014•16 citations

Factorizing the time evolution operator

P C García Quijas, L M Arévalo Aguilar

There is a widespread belief in the quantum physical community, and textbooks used to teach quantum mechanics, that it is a difficult task to apply the time evolution operator on an initial wavefunction. Because the Hamiltonian operator is, generally, the sum of two operators, then it is not possible to apply the time evolution operator on an initial wavefunction ψ( x , 0), for it implies using terms like . A possible solution is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wavefunction. However, the exponential operator does not directly factorize, i.e. . In this study we present a useful procedure for factorizing the time evolution operator when the argument of the exponential is a sum of two operators, which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like a particle subject to a constant force and a harmonic oscillator. Also, we discuss an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

arXiv:quant-ph/0603253•Physica Scripta•Jan 12, 2007•16 citations

Unitary approach to the quantum forced harmonic oscillator

D. Velasco-Martínez, Victor Guadalupe Ibarra‐Sierra, Juan Carlos Sandoval-Santana +2 more

In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.

arXiv:1409.0236•Aug 31, 2014•1 citation

The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator

C M Cheng, P. C. W. Fung

The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a 1 (t)J 4 +a 2 (t)J 0 +a 3 (t) J - where J + , J 0 and J - are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.

Journal of Physics A Mathematical and General•Nov 21, 1988•97 citations
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.
There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function, because it implies to apply terms like (A+B)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize. In this work we present useful ways to factorizing the time evolution operator when the argument of the exponential is a sum of two operators which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed in previous papers published in this Journal. Finally, we discuss the possible insight that can be learned using this approach in teaching Quantum Mechanics.
In their paper ‘Overcoming misconceptions in quantum mechanics with the time evolution operator’, García Quijas and Arévalo Aguilar (2007 Eur. J. Phys. 28 147) examined the time-dependent wave function of a particle in the one-dimensional harmonic oscillator potential using two different methods. The two wave functions that the authors obtained through the methods have different analytical expressions. The authors showed numerically that the two wave functions lead to the same probability density. When the real parts of the wave functions are compared, however, they are different in their details. That was puzzling because both wave functions are supposed to be solutions of the same time-dependent Schrödinger equation with the same initial condition. We point out that the two wave functions are actually identical. We show this analytically.
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.
There is a widespread belief in the quantum physical community, and textbooks used to teach quantum mechanics, that it is a difficult task to apply the time evolution operator on an initial wavefunction. Because the Hamiltonian operator is, generally, the sum of two operators, then it is not possible to apply the time evolution operator on an initial wavefunction ψ( x , 0), for it implies using terms like . A possible solution is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wavefunction. However, the exponential operator does not directly factorize, i.e. . In this study we present a useful procedure for factorizing the time evolution operator when the argument of the exponential is a sum of two operators, which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like a particle subject to a constant force and a harmonic oscillator. Also, we discuss an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.