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

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Table Of Integrals Series And Products

We review the field of cavity optomechanics, which explores the interaction between electromagnetic radiation and nano- or micromechanical motion This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiation pressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavity quantum optomechanics experiments In addition, we describe the perspectives for fundamental quantum physics and for possible applications of optomechanical devices

Cavity Optomechanics

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

It is shown, that for quantum systems the vectorfield associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrodinger and Heisenberg pictures. We illustrate these ambiguities in terms of simple examples.

Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics

Explicit expressions for most interesting quantum observable operators in optical tomography representation are found. A general formalism of the symbols of the operators is presented in the optical tomography representation. The symbols of the operators in the form of singular and regular generalized functions are found, and suggestions for their use in experimental data processing in quantum tomography are given.

Description and measurement of observables in the optical tomographic probability representation of quantum mechanics

In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schr\"odinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.

Time-dependent invariants and Green functions in the probability representation of quantum mechanics

Long ago in quantum mechanics a discussion appeared about the problem of opening a completely absorbing shutter on which a stream of particles of definite velocity was impinged. The solution of the problem was obtained in a form entirely analogous to the optical one of diffraction by a straight edge. The argument of the Fresnel integrals was time dependent, and thus the first part in the title of this paper. In this paper we reformulate the problem in Wigner distributions and tomographical probabilities. In the former case the probability in phase space is very simple but, as it takes positive and negative values, the interpretation is ambiguous, though it gives a classical limit that agrees entirely with our intuition. In the latter case we can start with our initial conditions in a given reference frame, but obtain our final solution in an arbitrary frame of reference.

Vladimir I. Man’ko

Papers

Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics

Description and measurement of observables in the optical tomographic probability representation of quantum mechanics

Time-dependent invariants and Green functions in the probability representation of quantum mechanics

Diffraction in time in terms of wigner distributions and tomographic probabilities

Nonstationary Casimir effect and oscillator energy level shift