Showing papers by "Vladimir I. Man’ko published in 1999"
TL;DR: In this article, the marginal distributions along rotated directions in the ( t, ω ) plane were used to obtain time and frequency information for non-stationary signals, and the rigorous probability interpretation of the marginal distribution avoided all interpretation ambiguities.
135 citations
TL;DR: In this article, the authors introduce the concept of the "polarized" distance, which distinguishes the orthogonal states with different energies, and give new inequalities for the known Hilbert-Schmidt distance between neighbouring states and express this distance in terms of the quasiprobability distributions and the normally ordered moments.
Abstract: We introduce the concept of the "polarized" distance, which distinguishes the orthogonal states with different energies. We also give new inequalities for the known Hilbert-Schmidt distance between neighbouring states and express this distance in terms of the quasiprobability distributions and the normally ordered moments. Besides, we discuss the distance problem in the framework of the recently proposed "classical-like" formulation of quantum mechanics, based on the sympletic tomography scheme. The examples of Fock's, coherent, "Schrodinger cats", squeezed, phase and thermal states are considered.
49 citations
TL;DR: In this article, the authors reformulate the problem in Wigner distributions and tomographical probabilities and show that the probability in phase space is very simple but, as it takes positive and negative values, the interpretation is ambiguous.
Abstract: 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.
46 citations
TL;DR: It is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam.
Abstract: An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
27 citations
TL;DR: In this paper, the relation between the density matrix obeying the von Neumann equation and the wave function obeying Schrodinger equation is discussed in connection with the superposition principle of quantum states.
Abstract: The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrodinger equation is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given, and its relation to the addition law of vectors in the Hilbert space of states and the role of a constant phase factor of the wave function is elucidated. The superposition law of density matrices, Wigner functions, and tomographic probabilities describing quantum states in the probability representation of quantum mechanics is studied. Examples of spin-1/2 and Schrodinger-cat states of the harmonic oscillator are discussed. The connection of the addition law with the entanglement problem is considered.
27 citations
TL;DR: In this article, an explicit expression for the transition probability distribution for the classical propagator in terms of path integral is derived, and the evolution equation in the Bargmann representation of the optical tomography approach is obtained.
Abstract: In the probability representation of the standard quantum mechanics, the explicit expression (and its quasiclassical van-Fleck approximation) for the “classical” propagator (transition probability distribution), which completely describes the quantum system's evolution, is found in terms of the quantum propagator. An expression for the “classical” propagator in terms of path integral is derived. Examples of free motion and harmonic oscillator are considered. The evolution equation in the Bargmann representation of the optical tomography approach is obtained.
23 citations
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TL;DR: In this article, an explicit expression for the transition probability distribution in terms of the quantum propagator was derived for the classical propagator in the context of the optical tomography approach.
Abstract: In the probability representation of the standard quantum mechanics, the explicit expression (and its quasiclassical van-Fleck approximation) for the ``classical'' propagator (transition probability distribution), which completely describes the quantum system's evolution, is found in terms of the quantum propagator. An expression for the ``classical'' propagator in terms of path integral is derived. Examples of free motion and harmonic oscillator are considered. The evolution equation in the Bargmann representation of the optical tomography approach is obtained.
19 citations
TL;DR: In this article, the authors use the fact that some linear Hamiltonian systems can be considered as "finite level" quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonians.
Abstract: We use the fact that some linear Hamiltonian systems can be considered as "finite level" quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonian systems.
16 citations
TL;DR: The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied in this paper, where a new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the ''probability representation'' of quantum states is given.
Abstract: The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied Alternatives to the Schr\"odinger evolution equation and to the energy level equation written for the positive probability distribution are discussed Instead of the transition probability amplitude (Feynman path integral) a transition probability is introduced
A new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the ``probability representation'' of quantum states is given An equation for the propagator in the new formulation of quantum mechanics is derived Some paradoxes of quantum mechanics are reconsidered
14 citations
24 Feb 1999
TL;DR: In this article, a new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the probability representation of quantum states is given, and an equation for the propagator in this formulation of quantum mechanics is derived.
Abstract: The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied. Alternatives to the Schrodinger evolution equation and to the energy level equation written for the positive probability distribution are discussed. Instead of the transition probability amplitude (Feynman path integral) a transition probability is introduced. A new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the “probability representation” of quantum states is given. An equation for the propagator in the new formulation of quantum mechanics is derived. Some paradoxes of quantum mechanics are reconsidered.
14 citations
TL;DR: Using the invertible map of Wigner functions onto positive probability distribution functions, this article considered an action of the Heisenberg-Weyl group on the space of distributions associated with an irreducible representation.
Abstract: Using the invertible map of Wigner functions onto positive-probability-distribution functions we consider an action of the Heisenberg–Weyl group on the space of distributions associated with an irreducible representation. We discuss also the connection between linear and nonlinear coherent states starting with noncanonical nonlinear maps within classical dynamics.
TL;DR: In this article, a new quantum-mechanical kinetic equation for excited states of a damped oscillator is obtained explicitly and the difference between the position probability distributions, which determine (in the new formulation of quantum mechanics) states of the dammed oscillator within the framework of the Caldirola-Kanai model, the kinetic equation with a collision term, and the nonlinear Kostin equation, is analyzed.
Abstract: Solutions to a new quantum-mechanical kinetic equation for excited states of a damped oscillator are obtained explicitly. The difference between the position probability distributions, which determine (in the new formulation of quantum mechanics) states of the damped oscillator within the framework of the Caldirola-Kanai model, the kinetic equation with a collision term, and the nonlinear Kostin equation, is analyzed.
TL;DR: In this article, the spectrum of light scattered from a Bose-Einstein condensate is studied in the limit of particle-number conservation, and a description in terms of deformed bosons is invoked and this leads to a deviation from the usual predict spectrum's shape as soon as the number of particles decreases.
TL;DR: In this paper, it was shown that the angular momentum of a two-mode oscillator is not conserved in the adiabatic limit for the model of a slowly varying frequency of one of the modes.
Abstract: For the exactly solvable model of a two-mode oscillator with slowly varying frequency of one of the modes it is demonstrated that such a quantum number as the angular momentum of the oscillator is not conserved in the adiabatic limit.
TL;DR: In this paper, new time-dependent integrals of motion were found in the explicit form for magnetic dipole precessing in a constant magnetic field, which describe the initial values of spin projections on the coordinate axes.
Abstract: New time-dependent integrals of motion are found in the explicit form for magnetic dipole precessing in a constant magnetic field. The integrals of motion do not commute with the Hamiltonian. The new integrals of motion describe the initial values of spin projections on the coordinate axes. Commutation relations of the new integrals of motion determine the Lie algebraSU(2).