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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Fractal properties of the quantum superpositions

Alexander V. Gorokhov, Alexander V. Shaikin

Published Apr 20, 2004• 0 citations
Publisher

Abstract

Examples of Berry’s and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum fractal constructions are considered. The behavior of fractal dimensions of these very special superpositions of basic states of a particle in a potential box and harmonic oscillator is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in “fractal” state and interacting with another oscillator are calculated.

DOI
10.1117/12.558821

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Examples of Berry’s and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum fractal constructions are considered. The behavior of fractal dimensions of these very special superpositions of basic states of a particle in a potential box and harmonic oscillator is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in “fractal” state and interacting with another oscillator are calculated.
Examples of Berrys and WBBZ - (D.Wojcik, I.Bialynicki-Birula and K.Zyczkowski) quantum constructions are considered. The behavior of dimensions of these objects is investigated in going from coordinate representation to momentum and coherent state representations. The mean value of quanta and dispersion for isolated quantum harmonic oscillator prepared in fractal state and interacting with another oscillator are calculated.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density ${P}_{t}(x)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}|\ensuremath{\Psi}(x,t){|}^{2}$ is shown not to change during the time evolution. We prove a universal relation Dt=1+Dx/2{D}_{t}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1+D}_{x}/2Dt​=1+Dx​/2 linking the dimensions of space cross sections Dx{D}_{x}Dx​ and time cross sections Dt{D}_{t}Dt​ of the fractal quantum carpets.
A review is given of the properties of two types of quantum states, which have great uncertainty in the coordinate and momentum.Both are obtained from the states of a harmonic oscillator by means of certain transformations.The first type is correlated states, they are obtained from coherent states with a help of the Bogolyubov transform.The variances of the coordinate and momentum of such a state depend on the Bogolyubov transform parameters and can, in general, take arbitrarily large values.Their specific values are determined by the physical processes with which the Bogolyubov transform is realized.A concrete example of such physical process is considered.Another type are stretched states.Formally, they arise when the n-partial state of a harmonic oscillator undergoes a transform associated with a scale transformation of the phase space.The dispersion of the coordinate and momentum of these states depends on the scale transformation parameter and can also take arbitrarily large values.Physically stretched states can be obtained by passing n-photon states through a quantum amplifier.The role of the scale transformation of the phase space is played by the gain of the quantum amplifier.
A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ \ensuremath{\surd}2 , where P is the parity operator. Such A arises naturally in the q\ensuremath{\rightarrow}-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-q2{\mathit{q}}^{2}q2${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, ${\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.
We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.