Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

Ibarra-Sierra, V. G.; Sandoval-Santana, J. C.; Cardoso, J. L.; Kunold, A.

doi:10.1016/j.aop.2015.07.022

Cited by 20 publications

(19 citation statements)

References 77 publications

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“…where C(a, q , ωt/2) is the even Mathieu function with a = 4ω 2 1 /ω 2 and q = −2ω 2 0 /ω 2 . Notice that this set of coefficients had been found before by us [34,35] and others. [40,41] Nevertheless, these coefficients themselves are insufficient because they only permit to write the evolution operator in the form U A (see Equation (4)) whereas the calculation of H e requires the evolution operator to be written in the form of U B (see Equation (5)).…”

Section: Example 1: Harmonic Oscillator With Time-dependent Frequencysupporting

confidence: 62%

“…[5,33] This approach is based on our earlier results on the use of Lie algebras to find the evolution operator of time-dependent Hamitlonians. [34,35] However, the method presented here has major improvements and differences with respect to our previous work. First, it does not require the prior knowledge of the transformation rules arising from the commutators between the algebra's elements.…”

Section: Introductionmentioning

confidence: 85%

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Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

et al. 2019

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Time-driven quantum systems are important in many different fields of physics as cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator or the effective Hamiltonian. Finding these operators usually requires very complex calculations that often involve some approximations. To perform this task, a systematic scheme that can be cast in the form of a symbolic computational algorithm is presented. It is suitable for periodic and non-periodic potentials and, for convoluted systems, can also be adapted to yield numerical solutions. The method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples are provided: harmonic oscillator with time-dependent frequency (Paul trap), modulated optical lattice, time-driven quantum oscillator, a step-wise driving of a free particle, and the non-periodic Caldirola-Kanai Hamiltonian. To the extent of the authors' knowledge, whereas the exact form of Paul trap's evolution operator is well known, its effective Hamiltonian was until now unknown. The remaining four examples accurately reproduce previous results.

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Section: Example 1: Harmonic Oscillator With Time-dependent Frequencysupporting

confidence: 62%

Section: Introductionmentioning

confidence: 85%

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

et al. 2019

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“…In this case a 2 = eE x , a 3 = eE y , a 6 = a 7 = mω 2 c /8, a 9 = a 10 = 1/2m, a 14 = −a 15 = ω c /2 and a 1 = a 4 = a 5 = a 8 = a 11 = a 12 = a 13 = 0 where ω c = eB/m is the cyclotron frequency. For the sake of simplicity we consider the case where E x , E y and B are constant although the more general case where these quantities are time-dependent can, in principle, be dealt with [61]. Substituting the previous parameters into the system of ordinary differential equations given by (55)- (69) we obtain the explicit form of the α parameters.…”

Section: Two-dimensional Charged Particle Subject To An In-plane mentioning

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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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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“…Albeverio and Mazzucchi considered Schrödinger equation with a time-dependent quadratic plus quartic Hamiltonian and using Feynman path integral representation [25]. Ibarra-Sierra et al used 2 Advances in High Energy Physics the Lie-algebraic technique and solved the time-dependent harmonic oscillator and the bidimensional charged particle in time-dependent electromagnetic fields [26]. In [27], linear invariants and the dynamical invariant method are used to obtain the exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

Study of Time Evolution for Approximation of Two-Body Spinless Salpeter Equation in Presence of Time-Dependent Interaction

Sobhani

Hassanabadi

2016

Advances in High Energy Physics

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We approximate the two-body spinless Salpeter equation with the one which is valid in heavy quarks limit. We consider the resulting semirelativistic equation in a time-dependent formulation. We use the Lewis-Riesenfeld dynamical invariant method and series solution to obtain the solutions of the differential equation. We have also done some calculations in order to derive the time evolution operator for the considered problem.

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Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

Cited by 20 publications

References 77 publications

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

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

Study of Time Evolution for Approximation of Two-Body Spinless Salpeter Equation in Presence of Time-Dependent Interaction

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