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

Cheng, C M; Fung, P. C. W.

doi:10.1088/0305-4470/21/22/015

Cited by 87 publications

(60 citation statements)

References 25 publications

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“…The key element behind the Lie algebraic approach is that the general form of the evolution operator of such type of Hamiltonian can be expressed as [55][56][57][58][59] …”

Section: The Lie Algebraic Approachmentioning

confidence: 99%

“…These commutation relations can be comprehended as a particular realization of (3) where the structure constants are related to the coefficients found in Table I . Also in this Table, In particular ĥ 6 ,ĥ 9 ,ĥ 12 or ĥ 7 ,ĥ 10 ,ĥ 13 form the SU (1, 1) Lie algebra that has been used to study KanaiCaldirola Hamiltonians through the Lie algebraic approach [27,58]. The sub-algebras ĥ 1 ,ĥ 2 ,ĥ 4 ,ĥ 6 ,ĥ 9 ,ĥ 12 or ĥ 1 ,ĥ 3 ,ĥ 5 ,ĥ 7 ,ĥ 10 ,ĥ 13 correspond to the generalised one-dimensional harmonic oscillator [1,35] along the x and y axis respectively.…”

Section: Generalized Two-dimensional Quadratic Hamiltoniansmentioning

confidence: 99%

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Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Sandoval-Santana

Ibarra-Sierra

Cardoso

et al. 2016

Journal of Mathematical Physics

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We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.2

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“…The key element behind the Lie algebraic approach is that the general form of the evolution operator of such type of Hamiltonian can be expressed as [55][56][57][58][59] …”

Section: The Lie Algebraic Approachmentioning

confidence: 99%

Section: Generalized Two-dimensional Quadratic Hamiltoniansmentioning

confidence: 99%

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Sandoval-Santana

Ibarra-Sierra

Cardoso

et al. 2016

Journal of Mathematical Physics

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“…For example, we can recall the work of Balasubramanian [12] who discussed the time evolution operator method with time dependent Hamiltonians. Also see reference [13].…”

Section: Discussionmentioning

confidence: 99%

“…, see reference [16]. That is, we have to pass the exponentials to the right in the right hand side of Equation (13). In this case:…”

Section: The Methodsmentioning

confidence: 99%

Factorizing the time evolution operator

Quijas

Aguilar

2007

Phys. Scr.

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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 e itĤ/ 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 ψ(x, 0), for it implies to apply terms like (â +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, i. e. eâ +b = eâeb. In this work 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 the particle subject to a constant force and the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

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“…(11) are obtained from these solutions by combining Eqs. (13) and (15) to (17). We write the solutions as…”

Section: The T O Functionsmentioning

confidence: 99%

The Schrödinger System

Nieto

Truax

2001

Annals of Physics

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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

Cited by 87 publications

References 25 publications

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Factorizing the time evolution operator

The Schrödinger System

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