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DIFFUSION PROCESSES AND COHERENT STATES

S. De Martino, Silvio De Siena, Fabrizio Illuminati

Published Jan 1, 1994• 10 citations
OpenAlex

Abstract

It is shown that uncertainty relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker-Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials. 1

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It is shown that uncertainty relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker-Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials. 1
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It is shown that the Bohm equations for the phase SSS and squared modulus ρρρ of the quantum mechanical wave function can be derived from the classical ensemble equations admiting an aditional momentum psp_sps​ of the form proportional to the osmotic velocity in the Nelson stochastic mechanics and using the variational principle with appropriate change of variables. The possibility to treat gradSSS and psp_sps​ as two parts of the momentum of quantum ensemble particles is considered from the view point of uncertainty relations of Robertson - Schroedinger type on the examples of the stochastic image of quantum mechanical canonical coherent and squeezed states.
We consider additional aspects of the recently derived “minimum uncertainty” (μ) wavelets. In particular, we show that they are fundamentally related to both the harmonic oscillator eigenstates and the canonical coherent states that play a fundamental role in quantum dynamics. In addition, we derive new raising and lowering operators that apply to the μ-wavelets. Finally, we explore in some detail the senses in which the μ-wavelets form complete sets that can be used in a variety of applications in quantum dynamics.