Wave function collapses and 1/n energy spectrum induced by a Coulomb potential in a one-dimensional flat band system

We investigate the bound state problem in a one-dimensional flat band system with a Coulomb potential.It is found that,in the presence of a Coulomb potential of type I(with three equal diagonal elements),similarly to that in the twodimensional case,the flat band could not survive.At the same time,the flat band states are transformed into localized states with a logarithmic singularity near the center position.In addition,the wave function near the origin would collapse for an arbitrarily weak Coulomb potential.Due to the wave function collapses,the eigen-energies for a shifted Coulomb potential depend sensitively on the cut-off parameter.For a Coulomb potential of type II,there exist infinite bound states that are generated from the flat band.Furthermore,when the bound state energy is very near the flat band,the energy is inversely proportional to the natural number,e.g.,E_(n)∝1/n,n=1,2,3,...It is expected that the 1/n energy spectrum could be observed experimentally in the near future.

Entropy and Irreversibility in Classical and Quantum Mechanics

Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schr?dinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.