Nonlinearity and constrained quantum motion (original) (raw)
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NATO Science Series II: Mathematics, Physics and Chemistry, 2006
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
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UDC 517.9 The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered. Von Neumann first applied the spectral theory of self-adjoint operators on Hilbert spaces to explain the radiation spectra of atoms and the related matter stability [2] (1926); Fock was the first to introduce the notion of many-particle Hilbert space, named a Fock space, and introduced related creation and annihilation operators acting on it [3] (1932); Weyl understood the fundamental role of the notion of symmetry in physics and developed a physics-oriented group theory; moreover, he showed the importance of different representations of classical matrix groups for physics and studied unitary representations of the Heisenberg-Weyl group related to creation and annihilation operators on a Fock space [4] (1931). At the end of the 20th century, new developments were due to Faddeev with co-workers (quantum inverse spectral theory transform [5], 1978); Drinfeld, Donaldson, and Witten (quantum groups and algebras, quantum topology, and quantum superanalysis [6-8], 1982-1994);
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In this article, we proposed a method for describing the evolution of quantum physical systems. We define the action integral on the functional space and the entropy of distribution of observable values on the set of quantum states. Dynamic of quantum system in this article is described as dynamical system represented by one parametric semi group which is extremal of this action integral. Evolution is a chain of change of distribution of observable values. In the closed system, it must increase entropy. Based on the notion of entropy of distribution energy, on the principle of maximum entropy production, we get a picture of evolution of closed quantum systems.
Quantum conditions on dynamics and control in open systems
The Journal of Chemical Physics, 2008
Quantum conditions on the control of dynamics of a system coupled to an environment are obtained. Specifically, consider a system initially in a system subspace H0 of dimensionality M0, which evolves to populate system subspaces H1, H2 of dimensionality M1, M2. Then there always exists an initial state in H0 that does not evolve into H2 if M0 > dM2, where 2 ≤ d ≤ (M0 + M1 + M2) 2 is the number of operators in the Kraus representation. Note, significantly, that the maximum d can be far smaller than the dimension of the bath. If this condition is not satisfied then dynamics from H0 that avoids H2 can only be attained physically under stringent conditions. An example from molecular dynamics and spectroscopy, i.e. donor to acceptor energy transfer, is provided.
Quantum mechanics for totally constrained dynamical systems and evolving hilbert spaces
International Journal of Theoretical Physics, 1996
We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with hamiltonian constraint that allow to include systems with an evolving Hilbert Space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical system with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case.
The Quantum-Classical Transition in Nonlinear Dynamical Systems
2000
Viewed as approximations to quantum mechanics, classical evolutions can violate the positive-semidefiniteness of the density matrix. The nature of this violation suggests a classification of dynamical systems based on classical-quantum correspondence; we show that this can be used to identify when environmental interaction (decoherence) will be unsuccessful in inducing the quantum-classical transition. In particular, the late-time Wigner function can become positive without any corresponding approach to classical dynamics. In the light of these results, we emphasize key issues relevant for experiments studying the quantum-classical transition.
Is the dynamics of open quantum systems always linear
We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.
Nonlinear Generalisation of Quantum Mechanics
2021
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.