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arXiv:quant-ph/9902079 · AIP conference proceedings

Conventional quantum mechanics without wave function and density matrix

Vladimir I. Man’ko

Published Jan 1, 1999• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.58229

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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.
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The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.
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.
The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.
Abstract The review of a new probability representation of quantum states is presented, where the states are described by conventional probability distribution functions. The invertible map of the probability distribution onto density operators in the Hilbert space is found using the introduced operators called a quantizer–dequantizer, which specify the invertible map of operators of quantum observables onto functions and a product of the operators onto an associative product (star product) of the functions. Examples of a quantum oscillator and a spin-1/2 particle are considered. The kinetic equations for probabilities, specifying the evolution of the states of a quantum system, which are equivalent to Schrödinger and von Neumann equations, are derived explicitly.