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arXiv:quant-ph/0005058 · Journal of Physics A Mathematical and General

The Pauli equation for probability distributions

S Mancini, O V Man'ko, V I Man'ko, P Tombesi

Published Apr 10, 2001• 42 citations
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Abstract

The tomographic-probability distribution for a measurable coordinate and spin projection is introduced to describe quantum states as an alternative to the density matrix. An analogue of the Pauli equation for the spin-½ particle is obtained for such a probability distribution instead of the usual equation for the wavefunction. Examples of the tomographic description of Landau levels and coherent states of a charged particle moving in a constant magnetic field are presented.

DOI
10.1088/0305-4470/34/16/314

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The tomographic-probability distribution for a measurable coordinate and spin projection is introduced to describe quantum states as an alternative to the density matrix. An analogue of the Pauli equation for the spin-½ particle is obtained for such a probability distribution instead of the usual equation for the wavefunction. Examples of the tomographic description of Landau levels and coherent states of a charged particle moving in a constant magnetic field are presented.
The problem of moving of a charged particle in electromagnetic field is considered in terms of tomographic probability representation. The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. The obtained results are generalized in the present work and are applied to determining the transition probabilities between Landau levels. Transition probabilities are calculated using the symplectic tomograms instead of the wave functions. The same method is used in obtaining the transition probabilities between the Landau levels possessed by a charge moving in varying electromagnetic field.
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.
The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle without any redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistributions W⃗(q,p)\vec W({\mathbf q},{\mathbf p})W(q,p), Q⃗(q,p)\vec Q({\mathbf q},{\mathbf p})Q​(q,p), P⃗(α⃗)\vec P(\vecα)P(α) are introduced. The evolution equations for proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. The quantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in proposed representations and evolution equations which are analogs of Pauli equation are obtained. The propagator of evolution equation in the case of homogeneous and stationary magnetic field in Landau gauge is found and the evolution of initial entangled superposition of lower Landau levels in the vector optical representation is considered. The system of linear quantum oscillator with spin in vector optical tomography representation is considered and the evolution of initial entangled superposition of two lower Fock states and spin-up, spin-down states is studied in this representation.
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.