The semiclassical approximation in quantum mechanics. A new approach

Багров, В. Г.; Belov, V. V.; Kondrat'eva, M. F.

doi:10.1007/bf01015121

Cited by 11 publications

(11 citation statements)

References 4 publications

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“…An alternative approach allowing to reduce the semiclassical quantum dynamics to a closed system of ODE is proposed by Bagrov with co-workers [30,31,32,33]. The phase-space trajectories appearing in [30,31,32,33] are connected to specific quantum states like in the de Broglie -Bohm theory.…”

Section: Composition Law For Quantum Trajectories and Energy Consementioning

confidence: 99%

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Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Krivoruchenko

Faessler

2007

Journal of Mathematical Physics

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The knowledge of quantum phase flow induced under the Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton's equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement "quantum trajectory belongs to a constraint submanifold" can be changed e.g. to the opposite by a unitary transformation. Some of relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves hamiltonian and constraint star-functions.

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Section: Composition Law For Quantum Trajectories and Energy Consementioning

confidence: 99%

“…The phase-space trajectories appearing in [30,31,32,33] are connected to specific quantum states like in the de Broglie -Bohm theory.…”

Section: Composition Law For Quantum Trajectories and Energy Consementioning

confidence: 99%

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Krivoruchenko

Faessler

2007

Journal of Mathematical Physics

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“…In the limit of → 0, the centroid of such a quasi-soliton moves in the phase space along the trajectory of this dynamic system: at each point in time, the semiclassically concentrated state is efficiently concentrated in the neighborhood of the point X(t, 0) (in the x representation) and in the neighborhood of the point P (t,0) (in the p representation). Note that a similar set of equations in quantum means has been obtained in [3,4] for the linear case (Schrödinger equation), and in [5] for a more general case. It has been shown [7,8] that these equations are Poisson equations with respect to the (degenerate) nonlinear Dirac bracket.…”

Section: − I∂ T +ᏼ T |ψ |mentioning

confidence: 96%

The trajectory‐coherent approximation and the system of moments for the Hartree type equation

Belov

Trifonov

Шаповалов

2002

International Journal of Mathematics and Mathematical Sciences

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101

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The general construction of semiclassically concentrated solutions to the Hartree type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ( → 0), are constructed with a power accuracy of O( N/2 ), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.2000 Mathematics Subject Classification: 81Q20, 35Q55, 47G20.

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“…This approach for the 1-dimensional Schro dinger equation was proposed in [28] and for the N-dimensional case N 1 in [29 31]. This system of ordinary equations has been called M-system 3 or the Hamilton Ehrenfest system.…”

Section: (K)mentioning

confidence: 98%

Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

1996

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The semiclassical approximation in quantum mechanics. A new approach

Cited by 11 publications

References 4 publications

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

The trajectory‐coherent approximation and the system of moments for the Hartree type equation

Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

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