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arXiv:cond-mat/0003289

Description of Quantum Beams Using a Stationary Incoherent Superposition of Wave Packets

Xavier Oriols, Jordi Suñé

Published Mar 16, 2000• 0 citations
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Abstract

An original entity is defined to study quantum beams of conserved particles. The beam is described as a Stationary Incoherent Superposition of Wave packets (SISOW). Mean values are time-independent confirming that the SISOW describes a stationary flux of particles. The SISOW is a general entity that contains the Hamiltonian eigenstates as a particular limit. The decomposition of the SISOW into wave packets provides intrinsic dynamic information which is not accessible from Hamiltonian eigenstates. The SISOW provides and adequate framework to study mesoscopic systems

DOI
10.48550/arxiv.cond-mat/0003289

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An original entity is defined to study quantum beams of conserved particles. The beam is described as a Stationary Incoherent Superposition of Wave packets (SISOW). Mean values are time-independent confirming that the SISOW describes a stationary flux of particles. The SISOW is a general entity that contains the Hamiltonian eigenstates as a particular limit. The decomposition of the SISOW into wave packets provides intrinsic dynamic information which is not accessible from Hamiltonian eigenstates. The SISOW provides and adequate framework to study mesoscopic systems
A beam of noninteracting particles, bosons or fermions, is described by the superposition of stochastic wave packets. This description allows in each case (bosons or fermions) the determination of the detection process of the particles. This process is defined by the set of the ppp-order coincidence probability densities, a general formulation of which is given. In the case of a stationary and weak incoherent beam, these coincidence probability densities are studied thoroughly and several results are obtained. The well-known bunching effect for bosons and the "antibunching effect" for fermions are shown to come from the detection of indistinguishable particles. In the boson case, all the well-known results for thermal light are found. In the fermion case, the detection process is, under certain conditions, identical to a renewal process.
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the
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The behavior of classical and beams in stationary media is shown to be ruled by a Wave Potential function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the case, the form of Bohm's Quantum Potential, which is therefore not so much a quantum as a wave property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
The behavior of classical and quantum wave beams in stationary media is shown to be ruled by a "Wave Potential" function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the quantum case, the form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the quantum case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
A consistent description of the motion of a particle by wave propagation implies not only a geometrical equivalence between a trajectory and a ray but also a kinematical equivalence between the time rates at which the particle or wave is traversing its path. This may be tentatively achieved by using a wave configuration, known as a wave packet, represented as a continuous superposition of monochromatic waves. If the wave packet can be regarded as a point, having zero amplitude everywhere except in a very small region, its motion along a trajectory might be interpreted as the motion of a single particle. The representation of a particle in space and time by means of a highly localized wave packet is a basic assumption of wave mechanics. The behavior of the wave packet as a whole will account for experiments where particle-like properties are observed, whereas individual wavelength components of packet will appear in experiments which depend upon interference and diffraction effects.