Fractal Wave Research Articles

Unusual quantum states: non–locality, entropy, Maxwell's demon and fractals

This paper analyses the mathematical properties of some unusual quantum states that are constructed by inserting an impenetrable barrier into a chamber confining a single particle. If the barrier is inserted at a fixed node of the wave function, then the energy of the system is unchanged. After barrier insertion, a measurement is made on one side of the chamber to determine if the particle is physically present. The measurement causes the wave function to collapse, and the energy contained in the subchamber where the particle is not present transfers instantaneously to the other subchamber where the particle now exists. This thought experiment constitutes an elementary example of an Einstein–Podolsky–Rosen experiment based on energy conservation rather than momentum or angular–momentum conservation. A more interesting situation arises when one inserts the barrier at a point that is not a fixed node of the wave function because this process changes the energy of the system; the faster the barrier is inserted, the greater the change in the energy. At the point of a sudden insertion the energy density becomes infinite; this energy instantly propagates across the subchamber and causes the wave function to become fractal. If an energy measurement is carried out on such a fractal wave function, the resulting mixed state has finite non–zero entropy. Fractal mixed states having unbounded entropy are also constructed and their properties are discussed. For a finite time insertion of the barrier, Landauer's principle is shown to be insufficient to resolve the apparent violation of the second law of thermodynamics that arises when a Maxwell demon is present. This problem is resolved by calculating the energy required to insert the barrier.

Chaos in different far-off cosmic rays: a fractal wave model

The air shower arrival time intervals (ASATIs) were observed for several years at five different far-off stations to study the chaotic feature of the cosmic rays. The total number of chaotic time series of ASATIs is 99 from February 1994 to November 2000. The chaotic ASATIs appear six times simultaneously between two different stations, showing a higher probability of simultaneous chaos detection than the random occurrence as implied by the fractal wave pattern of cosmic ray clouds. The average fractal dimension of the fractional chaotic ASATIs for each station is 3.7 ± 0.1 pm with average duration of 23 h. The distribution of the representative (central) right ascension of air showers among 99 chaotic ASATIs has two peaks around 4 h and 20 h. The time variation of the chaotic feature of the ASATIs for several days shows a quasi-periodic behaviour with the rotation of the earth. These results seem to indicate that the chaotic cosmic rays arrive at the earth, forming wave trains like a fractal pattern which covers an area of the galactic plane surrounding the earth.

Fractal characteristics of critical and localized states in incommensurate quantum systems

Incommensurate quantum systems with two competing periodicities exhibit metallic (with Bloch-type extended wave functions), insulating (with exponentially localized wave functions) as well ascritical (with fractal wave functions) phases. An exact renormalization method, which takes into account the inherent incommensurability, is used to obtain the phase diagram of various quantum models for the two-dimensional electron gas and for quantum spin chains in a magnetic field. In this approach, the scaling properties of the fractal eigenstates are characterized by a fixed point or a strange invariant set of the renormalization flow. One of our novel results is the existence of self-similar fluctuations in the localized states once the exponentially decaying envelope is factorized out. In almost all cases under investigation here, the universality classes can be broadly classified as those of the nearest-neighbor square or triangular lattices.

Renormalization approach to quasiperiodic tight binding models

Tight binding models such as the Harper equation exhibiting the localization transition are studied using a nonperturbative renormalization method. The self-similarity underlying the fractal wave function is described by an infinite set of universal scaling ratios. Our studies confirm the existence of a fat critical phase in perturbed Harper models describing quantum spin chains and shows the existence of two new universality classes. Furthermore, the decimation scheme proposed here provides a new method for finding the minimum and the maximum energy eigenvalues up to machine precision.

Renormalization approach to quasiperiodic tight binding models

Tight binding models such as the Harper equation exhibiting the localization transition are studied using a nonperturbative renormalization method. The self-similarity underlying the fractal wave function is described by an infinite set of universal scaling ratios. Our studies confirm the existence of a fat critical phase in perturbed Harper models describing quantum spin chains and shows the existence of two new universality classes. Furthermore, the decimation scheme proposed here provides a new method for finding the minimum and the maximum energy eigenvalues up to machine precision.

Coherent-state representation of quantum fractals in incommensurate systems

We study the fractal behavior of quantum eigenstates in incommensurate systems, described by the Harper's or almost-Mathieu equation. It is shown that the coherent-state representation (or, equivalently, the coarse-grained Wigner function) of the fractal wave functions reduces to classical periodic closed orbits in the semiclassical limit ℏ → 0.