arXiv:1212.6784

On Quantization, the Generalized Schrödinger Equation and Classical Mechanics

K. R. W. Jones

Published Dec 30, 2012• 0 citations

Abstract

Using a new state-dependent, $λ$-deformable, linear functional operator, ${\cal Q}_ψ^λ$, which presents a natural $C^{\infty}$ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case ${\cal Q}_ψ^{1}$ reproduces linear quantum mechanics, whereas ${\cal Q}_ψ^{0}$ admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.

DOI: 10.48550/arxiv.1212.6784

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arXiv:1212.6784

On Quantization, the Generalized Schrödinger Equation and Classical Mechanics

K. R. W. Jones

Published Dec 30, 2012• 0 citations

arXiv HTML PDF

Abstract

Using a new state-dependent,

λ

-deformable, linear functional operator,

{\cal Q}_ψ^λ

, which presents a natural

C^{\infty}

deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case

{\cal Q}_ψ^{1}

reproduces linear quantum mechanics, whereas

{\cal Q}_ψ^{0}

admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.

DOI: 10.48550/arxiv.1212.6784

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Quantum mechanics and classical trajectories

Christoph Nölle

The classical limit $\hbar$ ->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schrödinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schrödinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.The classical limit $\hbar$->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schrödinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schrödinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.

arXiv:1005.3786May 20, 20101 citation

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AIP conference proceedingsJan 1, 20114 citations

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Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type ${f}_{d}=k(t)v$ , where $k(t)$ is a scalar depending only on the time and $v$ is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as ${f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}$ , where $\mathcal{K}(\mathbf{x},t)$ is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type ${f}_{d}=k(t)v$, where $k(t)$ is a scalar depending only on the time and $v$ is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as ${f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}$, where $\mathcal{K}(\mathbf{x},t)$ is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.

Physical Review AFeb 2, 20183 citations

Using a new state-dependent,

λ

-deformable, linear functional operator,

{\cal Q}_ψ^λ

, which presents a natural

C^{\infty}

deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case

{\cal Q}_ψ^{1}

reproduces linear quantum mechanics, whereas

{\cal Q}_ψ^{0}

The classical limit

\hbar

->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schrödinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schrödinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.

\hbar

proves immaterial for the purpose of constructing this alternative representation of classical point mechanics. The new feature which

\hbar

does mediate concerns a simple embedding of the quantum geometric phase within classical mechanics. We discuss problems of physical interpretation and discover a simple route to recover the ordinary linear Schrödinger equation.

{f}_{d}=k(t)v

, where

k(t)

is a scalar depending only on the time and

v

is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as

{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}

, where

\mathcal{K}(\mathbf{x},t)

is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.