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arXiv:quant-ph/0311169

Reciprocal Schrödinger Equation: Durations of Delay and of Final States Formation in Processes of Scattering

Mark E. Perel’man

Published Nov 25, 2003• 0 citations
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Abstract

The reciprocal Schrödinger equation $\partial S(ω,{\bf r}% )/i\partial ω=\hatτ(ω,{\bf r}) S(ω,{\bf r})$ for $S$-matrix with temporal operator instead the Hamiltonian is established via the Legendre transformation of classical action function. Corresponding temporal functions are expressed via propagators of interacting fields. Their real parts $τ_{1}$are equivalent to the Wigner-Smith delay durations at process of scattering and imaginary parts $τ_{2}$ express the duration of final states formation (dressing). As an apparent example, they can be clearly interpreted in the oscillator model via polarization ($% τ_{1}$) and conductivity ($τ_{2}$) of medium. The $τ$-functions are interconnected by the dispersion relations of Kramers-Krönig type. From them follows, in particular, that $τ_{2}$ is twice bigger than the uncertainty value and thereby is measurable; it must be negative at some tunnel transitions and thus can explain the observed superluminal transfer of excitations at near field intervals (M.E.Perel'man. In: arXiv. physics/0309123). The covariant generalizations of reciprocal equation clarifies the adiabatic hypothesis of scattering theory as the requirement: $% τ_{2}\to 0$ at infinity future and elucidate the physical sense of some renormalization procedures.

DOI
10.48550/arxiv.quant-ph/0311169

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The reciprocal Schrödinger equation ∂S(ω,r\partial S(ω,{\bf r}% )/i\partial ω=\hatτ(ω,{\bf r}) S(ω,{\bf r})∂S(ω,r for SSS-matrix with temporal operator instead the Hamiltonian is established via the Legendre transformation of classical action function. Corresponding temporal functions are expressed via propagators of interacting fields. Their real parts τ1τ_{1}τ1​are equivalent to the Wigner-Smith delay durations at process of scattering and imaginary parts τ2τ_{2}τ2​ express the duration of final states formation (dressing). As an apparent example, they can be clearly interpreted in the oscillator model via polarization (% τ_{1}) and conductivity (τ2τ_{2}τ2​) of medium. The τττ-functions are interconnected by the dispersion relations of Kramers-Krönig type. From them follows, in particular, that τ2τ_{2}τ2​ is twice bigger than the uncertainty value and thereby is measurable; it must be negative at some tunnel transitions and thus can explain the observed superluminal transfer of excitations at near field intervals (M.E.Perel'man. In: arXiv. physics/0309123). The covariant generalizations of reciprocal equation clarifies the adiabatic hypothesis of scattering theory as the requirement: % τ_{2}\to 0 at infinity future and elucidate the physical sense of some renormalization procedures.
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Configuration space properties of the scattering operator and time delay for potentials decaying like ∣χ∣−α, α>1\vert \chi \vert^{-\alpha}, \, \alpha > 1∣χ∣−α,α>1
The time-delay in a simple quantum mechanical scattering process is defined to be the limit as r → ∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x)∼|x| #75α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1#75δ (δ > 0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation
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Specifically, when X(θ)=∑k=1nfk(θ)Xk{X{(\theta)}} = {\sum_{k = 1}^{n}{f_{k}{(\theta)}X_{k}}}X(θ)=∑k=1n​fk​(θ)Xk​ is a system observable dependent upon the Markovian θ\thetaθ, then, ∫Tr(ρ(t,θ)X(θ))dθ\int{Tr{({\rho{(t,\theta)}X{(\theta)}})}{d\theta}}∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of X(t,θ(t))=U(t)∗X(θ(t))U(t){X{(t,{\theta{(t)}})}} = {U{(t)}^{\ast}X{({\theta{(t)}})}U{(t)}}X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time tttThis paper explores some of the applications of quantum stochastic calculus (qsc) as founded by R.L.Hudson and K.R.Parthasarathy and the theory of the quantum filter established by V.P.Belavkin based on qsc which was simplified considerably by John Gough, Claude Kostler and others to quantum field theory. Specifically, we first derive some general versions of the Belavkin quantum filter for the case when the Hamiltonian and Lindblad operators in the Hudson-Parthasarathy noisy Schrodinger equation (HPSDE) are functions of a classical Markovian time varying parameter process whose statistics is determined by an infinitesimal generator. Using this generalized Belavkin filter, we show how we can estimate on real time basis not only the evolving mixed state of the system but also the probability density of the Markovian parameter. Our generalized filter enables us to obtain a function ρ(t,θ)\rho{(t,\theta)}ρ(t,θ) that represents an estimate of the state of the quantum system at time ttt as well as the probability density of the parameter θ\thetaθ at time ttt. Specifically, when X(θ)=∑k=1nfk(θ)Xk{X{(\theta)}} = {\sum_{k = 1}^{n}{f_{k}{(\theta)}X_{k}}}X(θ)=∑k=1n​fk​(θ)Xk​ is a system observable dependent upon the Markovian θ\thetaθ, then, ∫Tr(ρ(t,θ)X(θ))dθ\int{Tr{({\rho{(t,\theta)}X{(\theta)}})}{d\theta}}∫Tr(ρ(t,θ)X(θ))dθ would represent the conditional expectation of X(t,θ(t))=U(t)∗X(θ(t))U(t){X{(t,{\theta{(t)}})}} = {U{(t)}^{\ast}X{({\theta{(t)}})}U{(t)}}X(t,θ(t))=U(t)∗X(θ(t))U(t) given non-demolition measurements at time ttt. After that, we show how the Weyl-Moyal-Wigner transform can be used to formulate noisy quantum evolutions as well as Belavkin filter equations in a classical language wherein intepretations of quantum observables and states are given in terms of classical complex functions of positions and momenta. Marginals of the Wigner distribution of the state for example represent probabiity densities of positions and momenta but the Wigner distribution of the state does not represent any joint probability density of the positions and momenta, indeed the Heisenberg uncertainty does not allow any such interpretation. We then illustrate how a quantum mechanical theory taking into account quantum noise can be constructed in a natural way by perturbing the electromagnetic field with quantum super-symmetric noise introducing even and odd system operators. We explain how the interaction between the matter supercurrent and the metric superfield in supergravity theories can be perturbed by quantum noise by adding supersymmetric quantum noise to the matter supercurrent. Further, with the hope that Feynman path integrals can be used in quantum noisy field theories to evaluate transition amplitudes of scattering, we propose a Feynnman path integral approach to the solution of the HPQSDE when the system can be described by canonical position and momentum operators. We then describe quantum scattering in the presence of quantum noise in the HPQSDE framework. Here either one or both of the Hamiltonians contains noise and the corresponding joint unitary evolution of system and bath is described by a HPQSDE and we illustrate some methods for computing the scattering wave operators and analyzing their existence using standard quantum scattering inequalities. 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By estimating the Markovian paremeters using our generalized Belvakin filter on a real time basis, we are therefore able to generate a counter Hamiltonian via causing a simulated counter classical field using this estimated parameter to interact with the propagating quantum field and hence mitigate the effect of the interaction of the classical field with the quantum field. One of the novel results of this paper is the exact representation of the Weyl-Moyal-Wigner transform of products of quantum operators that includes Lie brackets between operators to all orders in Planck’s constant. We use this idea to obtain a classical representation of the quantum Boltzmann equation in terms of Wigner densities that are complex valued functions of position and momenta representing the system state. Future work on Wigner transforming the quantum Boltzmann equation for all r-particle marginal states of an N-particle system is in progress. Work on interpreting the Belavkin filter and quantum Boltzmann equation in the Wigner domain in terms of corrections to the classical Fujisaki-Kushner-Kallianpur stochastic filter and the classical Boltzmann equation is also in progress.
The time-delay in a simple quantum mechanical scattering process is a simple mechanical scattering process is defined to be the limit as r→∞ of the difference of the sojourn times of a scattering state and of the associated free state in a ball of radius r, with the particle under the influence of a potential W(x). For W(x) ~ |x| −α (α>1) at infinity, we use a smooth localization function with the tail growing like r 1−α (δ>0, depending on W) instead of the characteristic function of a ball of radius r and it is shown that the time-delay exists and satisfies the Eisenbud-Wigner relation.