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Theory of Probability and Its Applications

Approximation of the Evolution Operator by Expectations of Functionals of Sums of Independent Random Variables

Il'dar Abdullovich Ibragimov, Nataliya V. Smorodina, Михаил Михайлович Фаддеев

Published Jan 1, 2019• 0 citations
Publisher

Abstract

A method of probabilistic approximation of the operator $e^{-itH}$, where $H = -\frac{1}{2}\,\frac{d^2}{dx^2}+V(x), V\in L_\infty({\mathbf{R}})$, in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of sums of independent identically distributed random variables.

DOI
10.1137/s0040585x97t989362

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In Newtonian mechanics, a space-dependent potential V(x) is associated with both a change in energy in x and a conservative force -dV/dx. There is no notion of an interaction through a resonance, even though classically resonance situations exist. A quantum free particle is represented by the space and time two-dimensional probability distributions exp(ipx) and exp(-iEt) with x and t representing independent variables, even though on average there exists the relationship x= p/m t. In the case of the time-independent Schrodinger equation, one uses a potential V(x) taken from “classical” physics which is combined with the nonclassical exp(ipx) (i.e. with time removed). An immediate issue, we argue, is that V(x) implies different potential energy values at each x, while exp(ipx) → cos(px) describes probabilities at different x points associated with the same p. According to classical mechanics, p should change from one x point to another in a deterministic manner governed by V(x). Thus there exists a contradiction. We argue that the probabilistic form cos(px) remains intact, but that the interaction between an exp(ipx) and V(x) involves the formation of a probability resonance which involves other exp(ipx)s in the problem. In other words, V(x)=Sum over k Vk exp(ikx) suggests that the potential also behaves in a probabilistic manner and one may have Vkexp(ikx) a(p-k) exp(i (p-k) x) = Vka(p-k) exp(ipx). One may argue that Sum over k Vk expIkx) is simply a mathematical formalism (namely a Fourier series) without any physical manifestation. In this note, however, we argue against this and suggest that the “resonance” involved is linked with physical probability distributions which exist in both V(x) and external energy-supplying objects (such as a photon). According to the time-independent Schrodinger equation there are discrete energy levels associated with this differential equation, so an entity which may provide energy is required to lift a quantum bound particle from one energy level to the next. Electrons in an atom absorb photons to move to higher levels. Thus the photon is an energy-supplier and the fact that it has a spatial wavelength does not seem to directly enter into the solution of the time-dependent Schrodinger equation. If, however, a particle may only interact through a probability resonance, we argue that it is not only the potential V(x) which must physically be able to interact in a probabilistic way, but also any energy entities used to supply energy. Thus we argue that a photon must have a spatial distribution i..e wavelength to allow it to be part of the formation of spatial resonances in the particle - V(x) system which it in fact does. Thus we argue that the wavelength of a photon is experimental evidence for x-probability distribution resonances which occur between V(x) and a bound particle.

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In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.

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Probabilistic Approximation of the Evolution Operator

Il'dar Abdullovich Ibragimov, Nataliya V. Smorodina, Михаил Михайлович Фаддеев

Functional Analysis and Its Applications•Apr 1, 2018•8 citations
A method of probabilistic approximation of the operator e−itHe^{-itH}e−itH, where H=−12 d2dx2+V(x),V∈L∞(R)H = -\frac{1}{2}\,\frac{d^2}{dx^2}+V(x), V\in L_\infty({\mathbf{R}})H=−21​dx2d2​+V(x),V∈L∞​(R), in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of sums of independent identically distributed random variables.
In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.
We consider random linear continuous operators Ω→L(H,H)Ω\to \mathcal{L}(\mathcal{H}, \mathcal{H})Ω→L(H,H) on a Hilbert space H\mathcal{H}H. For example, such random operators may be random quantum channels. The Central Limit Theorem is known for the sums of i.i.d. random operators. Instead of the sum, there may be considered the composition of random exponents eAi/ne^{A_i/n}eAi​/n. We obtain the Central Limit Theorem in the Weak Operator Topology for centralized and normalized random exponents of i.i.d. linear continuous operators on a Hilbert space.
In Newtonian mechanics, a space-dependent potential V(x) is associated with both a change in energy in x and a conservative force -dV/dx. There is no notion of an interaction through a resonance, even though classically resonance situations exist. A quantum free particle is represented by the space and time two-dimensional probability distributions exp(ipx) and exp(-iEt) with x and t representing independent variables, even though on average there exists the relationship x= p/m t. In the case of the time-independent Schrodinger equation, one uses a potential V(x) taken from “classical” physics which is combined with the nonclassical exp(ipx) (i.e. with time removed). An immediate issue, we argue, is that V(x) implies different potential energy values at each x, while exp(ipx) → cos(px) describes probabilities at different x points associated with the same p. According to classical mechanics, p should change from one x point to another in a deterministic manner governed by V(x). Thus there exists a contradiction. We argue that the probabilistic form cos(px) remains intact, but that the interaction between an exp(ipx) and V(x) involves the formation of a probability resonance which involves other exp(ipx)s in the problem. In other words, V(x)=Sum over k Vk exp(ikx) suggests that the potential also behaves in a probabilistic manner and one may have Vkexp(ikx) a(p-k) exp(i (p-k) x) = Vka(p-k) exp(ipx). One may argue that Sum over k Vk expIkx) is simply a mathematical formalism (namely a Fourier series) without any physical manifestation. In this note, however, we argue against this and suggest that the “resonance” involved is linked with physical probability distributions which exist in both V(x) and external energy-supplying objects (such as a photon). According to the time-independent Schrodinger equation there are discrete energy levels associated with this differential equation, so an entity which may provide energy is required to lift a quantum bound particle from one energy level to the next. Electrons in an atom absorb photons to move to higher levels. Thus the photon is an energy-supplier and the fact that it has a spatial wavelength does not seem to directly enter into the solution of the time-dependent Schrodinger equation. If, however, a particle may only interact through a probability resonance, we argue that it is not only the potential V(x) which must physically be able to interact in a probabilistic way, but also any energy entities used to supply energy. Thus we argue that a photon must have a spatial distribution i..e wavelength to allow it to be part of the formation of spatial resonances in the particle - V(x) system which it in fact does. Thus we argue that the wavelength of a photon is experimental evidence for x-probability distribution resonances which occur between V(x) and a bound particle.
In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.