Dynamical invariants for variable quadratic Hamiltonians

Suslov, Sergeĭ K.

doi:10.1088/0031-8949/81/05/055006

Cited by 31 publications

(68 citation statements)

References 84 publications

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“…[54] (see also [10], [15], [59] and the references therein for important special cases). An application to the electromagnetic-field quantization and a generalization of the coherent states are discussed in Refs.…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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Abstract. We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.The purpose of this Letter is to elaborate on a "missing" class of solutions to the time-dependent Schrödinger equation for the simple harmonic oscillator in one dimension. We also provide an interesting computer-animated feature of these solutions -the phase space oscillations of the electron density and the corresponding probability distribution of the particle linear momentum. As a result, a dynamic visualization of the fundamental Heisenberg Uncertainty Principle [27] is given [45], [62]. Symmetry and Hidden SolutionsThe time-dependent Schrödinger equation for the simple harmonic oscillator,has the following six-parameter family of square integrable solutionswhere H n (x) are the Hermite polynomials [51] and On the other hand, the "dynamic harmonic oscillator states" (1.2)-(1.9) are eigenfunctions,of the time-dependent quadratic invariant,with the required operator identity [15], [54]:(1.12)Here, the time-dependent annihilation a (t) and creation a † (t) operators are explicitly given bywith p = i −1 ∂/∂x in terms of our solutions (1.4)-(1.9). These operators satisfy the canonical commutation relation, 14) and the oscillator-type spectrum (1.10) of the dynamic invariant E can be obtained in a standard way by using the Heisenberg-Weyl algebra of the rasing and lowering operators (a "second quantization" [1], [42], the Fock states): Here, ψ n (x, t) = e i(2n+1)γ(t) Ψ n (x, t) (1.16) is the relation to the wave functions (1.2) with ϕ n (t) = − (2n + 1) [68] and the references therein) have attracted substantial attention over the years because of their great importance in many advanced quantum problems. Examples are coherent and squeezed states, uncertainty relations, Berry's phase, quantization of mechanical systems and Hamiltonian cosmology. More applications include, but are not limited to charged particle traps and motion in uniform magnetic fields, molecular spectroscopy and polyatomic molecules in varying external fields, crystals through which an electron is passing and exciting the oscillator modes, and other mode interactions with external fields. Quadratic Hamiltonians have particular applications in quantum electrodynamics because the electromagnetic field can be represented as a set of generalized driven harmonic oscillators [13], [20].The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresp...

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Section: −(β(T)x+ε(t))mentioning

confidence: 99%

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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“…For this, we employ the method of integrals of motion [20][21][22][23]. We consider a classical analog of the Hamiltonian (1)…”

Section: Green Function Of Driven Harmonic Oscillatormentioning

confidence: 99%

“…The coherent states of the oscillator and dynamic invariants [20] of such a system were found in [21][22][23]. On the other hand, a detail consideration of this problem in the tomographic-probability representation has never been performed.…”

Section: Introductionmentioning

confidence: 99%

The driven-oscillator evolution in the tomographic-probability representation

Lemeshevskiy

Man’ko

2012

J Russ Laser Res

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We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system's phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.

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“…The linear time-dependent invariants of multidimensional quadratic Hamiltonians in the position and momentum operators have been the subject of many research studies [1][2][3][4][5]. These constants of motion are useful to determine the propagators of the Hamiltonian systems and thus to study time-dependent problems in quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

Evolution and Entanglement of Gaussian States in the Parametric Amplifier

et al. 2016

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The linear time-dependent constants of motion of the parametric amplifier are obtained and used to determine in the tomographic-probability representation the evolution of a general two-mode Gaussian state. By means of the discretization of the continuous variable density matrix, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier. The obtained results for the nonlocal correlations are compared with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. This qubit portrait procedure is used to establish Bell-type's inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. Other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.

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Dynamical invariants for variable quadratic Hamiltonians

Cited by 31 publications

References 84 publications

On a hidden symmetry of quantum harmonic oscillators

On a hidden symmetry of quantum harmonic oscillators

The driven-oscillator evolution in the tomographic-probability representation

Evolution and Entanglement of Gaussian States in the Parametric Amplifier

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