Hamiltonian systems of equations for quantum means

Belov, V. V.; Kondrat'eva, M. F.

doi:10.1007/bf02266690

Cited by 10 publications

(11 citation statements)

References 8 publications

(10 reference statements)

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“…The approach is based on finite-dimensional approximation to this system for the case in which the components of 7/ satisfy the estimates #~,~ = 0(h(l~l+l~l) [3]. The aim of our study is to generalize these results to matrix Hamiltonians (K > 1).…”

Section: Here ~ and # Are Multi-inmces With Nonnegative Integer Compomentioning

confidence: 96%

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

Belov

Kondrat'eva

1995

Math Notes

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ABSTRACT. An infinite system of ordinary differential equations for ~,/~, and for averages of a set of operators is derived for quantum-mechanical problems with a (K x K) matrix Hamiltonian 74(~,/~), z E R N 9 The set of operators is chosen to be basis in the space MatKC @ UOSYN) , where U(WN) is the universal enveloping algebra of the Heisenberg-Weyl algebra FVN, generated by the time-dependent operators ], ~ -e(t) 9 I, and /~ -/~(t) 9 ], where ] is the identity operator and ~ and /~ are the averages of the position and momentum operators. The system in question can be written in Hamiltonian form; the corresponding Poisson bracket is degenerate and is equal to the sum of the standard bracket on R ~N with respect to the variables (~',/~) and the generalized Dirac bracket with respect to the other variables. The possibility of obtaining finite-dimensional approximations to the infinite-dimensional system in the semiclassical limit h ---, 0 is investigated.

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Section: Here ~ and # Are Multi-inmces With Nonnegative Integer Compomentioning

confidence: 96%

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

Belov

Kondrat'eva

1995

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“…The quantity on the left-hand side of (1.15) is an integral of Hamilton-Ehrenfest system (1.13) (see [20]). Consequently, it suffices that the uncertainty relation holds at the initial moment of time.…”

Section: The Hamilton-ehrenfest Systemmentioning

confidence: 99%

Symmetry operators of a Hartree-type equation with quadratic potential

2005

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We study the symmetry properties of a nonstationary one-dimensional Hartree-type equation with quadratic periodic potential and nonlocal nonlinearity. We find an explicit form of a nonlinear evolution operator for this equation and obtain a solution to a Cauchy problem in the class of semiclassically concentrated functions. We find parametric families of nonlinear symmetry operators of a Hartree-type equation (keeping invariant the set of solutions to this equation). Using the symmetry operators, we construct families of exact solutions to the equation. This approach constructively extends the ideas and methods of group analysis to the case of nonlinear integro-differential equations.

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“…The left-hand side of (1.5) is the integral of motion of the HES (1.3) (see [27]). Hence, it suffices that the uncertainty relation be fulfilled at the initial time.…”

Section: The Hamilton-ehrenfest Systemmentioning

confidence: 99%

Berry phases for the nonlocal Gross–Pitaevskii equation with a quadratic potential

Litvinets

Шаповалов

Trifonov

2006

J. Phys. A: Math. Gen.

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A countable set of asymptotic space -localized solutions is constructed by the complex germ method in the adiabatic approximation for the nonstationary Gross -Pitaevskii equation with nonlocal nonlinearity and a quadratic potential. The asymptotic parameter is 1/T , where T ≫ 1 is the adiabatic evolution time.A generalization of the Berry phase of the linear Schrödinger equation is formulated for the GrossPitaevskii equation. For the solutions constructed, the Berry phases are found in explicit form.

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Hamiltonian systems of equations for quantum means

Cited by 10 publications

References 8 publications

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

Symmetry operators of a Hartree-type equation with quadratic potential

Berry phases for the nonlocal Gross–Pitaevskii equation with a quadratic potential

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