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arXiv:quant-ph/0612059

Some thoughts on an information-theoretic motivated nonlinear Schrodinger equation

Rajesh R. Parwani

Published Dec 8, 2006• 0 citations
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Abstract

I begin by reviewing the arguments leading to a nonlinear generalisation of Schrodinger's equation within the context of the maximum uncertainty principle. Some exact and perturbative properties of that equation are then summarised: those results depend on a free regulating/interpolation parameter $η$. I discuss here how one may fix that parameter using energetics. Other issues discussed are, a linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation, and possible symmetry enhancements in the nonlinear theory.

DOI
10.48550/arxiv.quant-ph/0612059

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I begin by reviewing the arguments leading to a nonlinear generalisation of Schrodinger's equation within the context of the maximum uncertainty principle. Some exact and perturbative properties of that equation are then summarised: those results depend on a free regulating/interpolation parameter ηηη. I discuss here how one may fix that parameter using energetics. Other issues discussed are, a linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation, and possible symmetry enhancements in the nonlinear theory.
We update our understanding of nonlinear Schrödinger equations motivated through information theory. In particular we show that a q —deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest q = 1 case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value η = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, η might be encoding relativistic effects.
About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.
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.
A new theory for the description of dissipative systems by nonlinear Schrödinger-type field equations (NLSE’s) with logarithmic nonlinearity, which has been recently developed by the authors, is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems. Wave-packet solutions as well as time-dependent wave-function solutions are derived and discussed. In the limit of vanishing friction (friction constant γ→0) these solutions turn into the well-known solutions of the respective linear Schrödinger field equation. The same applies to the mean values of position, momentum, and energy, as well as to the uncertainty product of position and momentum. For γ≠0, however, interesting new effects appear. In contrast to the linear theory the uncertainty product of position and momentum does not diverge any more for infinitely long times, t→∞, but asymptotically approaches a definite constant value which depends on characteristic parameters of the system like its mass, initial width, and friction constant γ. Another effect, the faster spreading of the Gaussian wave-packet solution compared to the linear theory, can be explained with the help of a special property of our nonlinear differential equation. In a way similar to what is usually only known for linear differential equations, the wave-packet solutions of our NLSE can be obtained by superposition of the wave functions which are individually also solutions of the same NLSE. The properties of the time-dependent superposition coefficients appearing in this connection are discussed. The extension to the corresponding three-dimensional problem as well as the differences arising in the investigation of the NLSE’s of the free fall and the motion in a constant electric field are given. Concluding, some differences are discussed which appear applying our nonlinear field theory to describe periodic or aperiodic motions, respectively.