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International Journal of Quantum Chemistry

Coupled quantum‐classical description in the Schrödinger representation

Yu. A. Rylov

Published Jan 1, 1999• 0 citations
Publisher

Abstract

It is shown that the number of dynamical variables (computer storage) which is necessary for a description is similar for the coupled quantum-classical description and for the pure quantum one. In the Schrödinger representation, the dynamic equations of the coupled quantum-classical description are shown to be nonlinear in terms of the wave function and incompatible with the quantum axiomatics. As a result, the quantum-classical description appears to be impossible in the Heisenberg representation, because it uses essentially the quantum axiomatics, which is incompatible with the quantum-classical description. An idea of a description, intermediate between the purely quantum description and the classical one, is suggested. In this case, the number of dynamic variables (computer storage) is several times as large as the classical description, but it is much less than at the pure quantum description. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 483–498, 1999

DOI
10.1002/(sici)1097-461x(1999)72:5<483::aid-qua2>3.0.co;2-#

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It is shown that the number of dynamical variables (computer storage) which is necessary for a description is similar for the coupled quantum-classical description and for the pure quantum one. In the Schrödinger representation, the dynamic equations of the coupled quantum-classical description are shown to be nonlinear in terms of the wave function and incompatible with the quantum axiomatics. As a result, the quantum-classical description appears to be impossible in the Heisenberg representation, because it uses essentially the quantum axiomatics, which is incompatible with the quantum-classical description. An idea of a description, intermediate between the purely quantum description and the classical one, is suggested. In this case, the number of dynamic variables (computer storage) is several times as large as the classical description, but it is much less than at the pure quantum description. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 483–498, 1999
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Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation prin­ciple a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
New excitation operators which have perfect bosonic symmetry are constructed for many-electron wave functions by regarding the system of many electrons as that of many species of bosons. Any electronic configurations can be generated by the new bosonic “void” operators. A coherent state is constructed with the bosonic operators and is adopted as a trial function for the time-dependent variational principle. The equation of motion which has exactly the same form as Hamilton's equation in classical mechanics is obtained with the complex variational parameters, the number of which is equal to the number of electrons. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 71–75, 1998
Abstract Relations between the extended wave functions defined by Carbó‐Dorca in connection with studies on quantum similarity problems and the wave functions of Schrödinger, Lévy‐Leblond, and Dirac equations are investigated. In particular, it is pointed out that solutions of the Lévy‐Leblond equation are equal to the appropriate extensions of the corresponding Schrödinger wave functions. The quantum mechanical continuity equation is applied to analyze the physical meaning of the extended wave functions. Finally, usefulness of the concept of the extended wave function in designing relativistic variational calculations in kinetically balanced basis sets is briefly discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 331–337, 2001
Dynamical chaos in quantum systems (quantum chaos) has been studied by a large number of physicists [1-7]. It has been shown that quantum chaos is possible only in the quasiclassical region, for example in the case of quantum states near the dissociation limit in molecules or near the ionization limit in atoms and molecules. However, the quantum equations of motion are linear and hence they cannot contain dynamical chaos. In the present paper we discuss the solution of quantum problems which are nonlinear in the classical limit. It will be show that in solving the quantum problem in this case it is necessary to use the solution of the nonlinear classical problem forming the classical limit of the quantum problem. i. We note first that here we are not interested in the usual field nonlinearities considered, for example, in quantum optics. This is because field nonlinearities cannot determine the bifurcation properties of the quantum equations of motion since the latter are linear in the wave function ~ or density matrix p. In other words, dynamical chaos, which is intimately connected with the bifurcation properties of the corresponding equations, is not determined by nonlinearities in the fields in quantum equations of motion linear in ~ or 0o