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Physics of Atomic Nuclei

Classical-like description of quantum states and propagator for particles in time-independent and dispersing δ potentials

Vladimir I. Man’ko, Alexander S. Chikhachev

Published Aug 1, 2001• 4 citations
Publisher

Abstract

Probability distributions that determine completely the bound state in the problem of a quantum particle in one or two delta-function wells are derived within the recently developed tomographic representation of quantum states, where a state is characterized by a positive probability distribution. The quantum propagator for the Schrödinger equation is obtained for the problem of two dispersing delta-function wells.

DOI
10.1134/1.1398938

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Probability distributions that determine completely the bound state in the problem of a quantum particle in one or two delta-function wells are derived within the recently developed tomographic representation of quantum states, where a state is characterized by a positive probability distribution. The quantum propagator for the Schrödinger equation is obtained for the problem of two dispersing delta-function wells.
In connection with the International Year of Quantum Science and Technology, a review of joint works of the Lebedev Institute and different Mexican research groups is presented; specially related to solving the old problem of the state description not only by wave functions but also by conventional probability distributions analogous to quasiprobability distributions like the Wigner function. Also, explicit expressions of tomographic representations describing the quantum states of particles moving in known potential wells are obtained and briefly discussed. In particular, we present the examples of the tomographic distributions for the free evolution, finite and infinite potential wells, and the Morse potential. Additional to this, an extension of the Peres--Horodecki separability criteria for momentum probability distributions is presented in the case of bipartite, asymmetrical, real states.
The quantizer–dequantizer method is employed. Using the construction of probability distributions describing density operators of a quantum system states, the connection between the Feynman path integral and the time evolution of the density operator (Landau density matrix) as well as the wave function of the stateconsidered. For single–mode systems with continuous variables, a tomographic propagator is introduced in the probability representation of quantum mechanics. An explicit expression for the probability in terms of the Green function of the Schrödinger equation is obtained. Equations for the Green functions defined by arbitrary integrals of motion are derived. Examples of probability distributions describing the evolution of state of a free particle, as well as states of systems with integrals of motion that depend on time (oscillator type) are discussed.
Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrödinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.
The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied. Alternatives to the Schrödinger 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.