There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Mathematical Handbook for Scientists and Engineers

A tutorial on modeling and analysis of dynamic social networks. Part I

Often quantum systems are not isolated and interactions with their environments must be taken into account. In such open quantum systems these environmental interactions can lead to decoherence and dissipation, which have a marked influence on the properties of the quantum system. In many instances the environment is well-approximated by classical mechanics, so that one is led to consider the dynamics of open quantum-classical systems. Since a full quantum dynamical description of large many-body systems is not currently feasible, mixed quantum-classical methods can provide accurate and computationally tractable ways to follow the dynamics of both the system and its environment. This review focuses on quantum-classical Liouville dynamics, one of several quantum-classical descriptions, and discusses the problems that arise when one attempts to combine quantum and classical mechanics, coherence and decoherence in quantum-classical systems, nonadiabatic dynamics, surface-hopping and mean-field theories and their relation to quantum-classical Liouville dynamics, as well as methods for simulating the dynamics.

Quantum Dynamics in Open Quantum-Classical Systems

Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential

The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum Sp information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a, while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ(x) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. The Bialynicki–Birula–Mycielski (BBM) inequality is also tested for these states and found to hold.

Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well

We generalize the Dirac equation to D+1 space–time. The conserved angular momentum operators and their quantum numbers are discussed. The eigenfunctions of the total angular momentums are calculated for both odd D and even D cases. The exact solutions of the D+1-dimensional radial equations of the Dirac equation with a Coulomb plus scalar potential are analytically presented by studying the Tricomi equations obtained from a pair of coupled first-order ones. The eigenvalues are also discussed in some detail.

http://fiz.upt.ro/articole/126074128JMP%2044_10(2003).pdf

Group theory approach to the Dirac equation with a Coulomb plus scalar potential in D+1 dimensions

In the paper we have investigated some properties of the Barut-Giraradello coherent states (BGCS) (the eigenstates of the SU(1,1) lowering generator K-) for the Hamiltonian of the pseudoharmonic oscillator (PHO) By using these states, the diagonal P-representation of the density operator is constructed as a new result for this potential In addition, we have calculated some thermal expectation values for the quantum canonical gas of the PHOs The calculations using the BGCS representation seem to be much simpler and easier to program than the corresponding calculations in other representations (eg the position representation) At the end of the paper the time dependence of these states is presented

Barut-Girardello coherent states of the pseudoharmonic oscillator

http://fiz.upt.ro/articole/126074128Phys%20Lett%20A%20316(2003)369-381.pdf

Gazeau–Klauder quasi-coherent states for the Morse oscillator

The comprehensive conditions concerning the harmonic limit of the Morse oscillator are investigated in this paper. These conditions extend the over-simple "classical" condition, which results by truncating the expansion of the Morse potential function in the power series of the coordinate close to the equilibrium position. The validity of the obtained results is tested by calculating the harmonic limit of the expected values of some characteristic physical observables of the Morse oscillator systems.

Dušan Popov

Papers

Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well

Group theory approach to the Dirac equation with a Coulomb plus scalar potential in D+1 dimensions

Barut-Girardello coherent states of the pseudoharmonic oscillator

Gazeau–Klauder quasi-coherent states for the Morse oscillator

Considerations Concerning the Harmonic Limit of the Morse Oscillator