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arXiv:1312.4195

On an interpolative Schrödinger equation and an alternative classical limit

K. R. W. Jones

Published Dec 15, 2013• 0 citations
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Abstract

We introduce a simple deformed quantization prescription that interpolates the classical and quantum sectors of Weinberg's nonlinear quantum theory. The result is a novel classical limit where $\hbar$ is kept fixed while a dimensionless mesoscopic parameter, $λ\in[0,1]$, goes to zero. Unlike the standard classical limit, which holds good up to a certain timescale, ours is a precise limit incorporating true dynamical chaos, no dispersion, an absence of macroscopic superpositions and a complete recovery of the symplectic geometry of classical phase space. We develop the formalism, and discover that energy levels suffer a {\em generic perturbation\/}. Exactly, they become $E(λ^{2}\hbar)$, where $λ= 1$ gives the standard prediction. Exact interpolative eigenstates can be similarly constructed. Unlike the linear case, these need no longer be orthogonal. A formal solution for the interpolative dynamics is given, and we exhibit the free particle as one exactly soluble case. Dispersion is reduced, to vanish at $λ= 0$. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation.

DOI
10.48550/arxiv.1312.4195

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We introduce a simple deformed quantization prescription that interpolates the classical and quantum sectors of Weinberg's nonlinear quantum theory. The result is a novel classical limit where ℏ\hbarℏ is kept fixed while a dimensionless mesoscopic parameter, λ∈[0,1]λ\in[0,1]λ∈[0,1], goes to zero. Unlike the standard classical limit, which holds good up to a certain timescale, ours is a precise limit incorporating true dynamical chaos, no dispersion, an absence of macroscopic superpositions and a complete recovery of the symplectic geometry of classical phase space. We develop the formalism, and discover that energy levels suffer a {\em generic perturbation/}. Exactly, they become E(λ2ℏ)E(λ^{2}\hbar)E(λ2ℏ), where λ=1λ= 1λ=1 gives the standard prediction. Exact interpolative eigenstates can be similarly constructed. Unlike the linear case, these need no longer be orthogonal. A formal solution for the interpolative dynamics is given, and we exhibit the free particle as one exactly soluble case. Dispersion is reduced, to vanish at λ=0λ= 0λ=0. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation.
Using Weinberg's generalized nonlinear quantum dynamics [Ann. Phys. (N.Y.) 194, 336 (1989)], we propose a simple way to form an interpolative dynamical system that joins the classical and quantum regimes. By altering a dimensionless control parameter \ensuremath{\lambda}\ensuremath{\in}[0,1], one can smoothly metamorphose quantum evolution into classical evolution, keeping the Hilbert-space dimension and physical constants unchanged. The method suggests an approach to studies of dynamical chaos suppression in quantized classically chaotic systems.
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We theoretically investigate the stability and linear oscillatory behavior of a naturally unstable particle whose potential energy is harmonically modulated. We find this fundamental dynamical system is analogous in time to a quantum harmonic oscillator. In a certain modulation limit, a.k.a. the Kapitza regime, the modulated oscillator can behave like an effective classic harmonic oscillator. But in the overlooked opposite limit, the stable modes of vibrations are quantized in the modulation parameter space. By analogy with the statistical interpretation of quantum physics, those modes can be characterized by the time-energy uncertainty relation of a quantum harmonic oscillator. Reducing the almost-periodic vibrational modes of the particle to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrodinger stationary wave equation with a harmonic potential. This reduction process introduces two features reminiscent of the quantum realm: a wave-particle duality and a loss of causality that could legitimate a statistical interpretation of the computed eigenfunctions. These results shed new light on periodically time-varying linear dynamical systems and open an original path in the recently revived field of quantum mechanical analogs.
We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.