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

arXiv:quant-ph/0603253 · Physica Scripta

Factorizing the time evolution operator

P C García Quijas, L M Arévalo Aguilar

Published Jan 12, 2007• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0031-8949/75/2/012

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

Overcoming misconceptions in quantum mechanics with the time evolution operator

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

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.

European Journal of Physics•Jan 2, 2007•27 citations

Total time derivatives of operators in elementary quantum mechanics

Mark Andrews

The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator.

American Journal of Physics•Mar 11, 2003•6 citations

Operator Methods for the Time Evolution of Wave Functions

K. P. Michnicki

A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.

arXiv:0811.3431•Nov 20, 2008•0 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

Wave mechanics

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.

Cambridge University Press eBooks•Mar 11, 2004•0 citations
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.
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.
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.
The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator.
A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.
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.
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.