211 research outputs found
Bayesian Inverse Quantum Theory
A Bayesian approach is developed to determine quantum mechanical potentials
from empirical data. Bayesian methods, combining empirical measurements and "a
priori" information, provide flexible tools for such empirical learning
problems. The paper presents the basic theory, concentrating in particular on
measurements of particle coordinates in quantum mechanical systems at finite
temperature. The computational feasibility of the approach is demonstrated by
numerical case studies. Finally, it is shown how the approach can be
generalized to such many-body and few-body systems for which a mean field
description is appropriate. This is done by means of a Bayesian inverse
Hartree-Fock approximation.Comment: LaTex, 32 pages, 19 figure
Bayesian Reconstruction of Approximately Periodic Potentials at Finite Temperature
The paper discusses the reconstruction of potentials for quantum systems at
finite temperatures from observational data. A nonparametric approach is
developed, based on the framework of Bayesian statistics, to solve such inverse
problems. Besides the specific model of quantum statistics giving the
probability of observational data, a Bayesian approach is essentially based on
"a priori" information available for the potential. Different possibilities to
implement "a priori" information are discussed in detail, including
hyperparameters, hyperfields, and non--Gaussian auxiliary fields. Special
emphasis is put on the reconstruction of potentials with approximate
periodicity. The feasibility of the approach is demonstrated for a numerical
model.Comment: 18 pages, 17 figures, LaTe
Mean Field Methods for Atomic and Nuclear Reactions: The Link between Time--Dependent and Time--Independent Approaches
Three variants of mean field methods for atomic and nuclear reactions are
compared with respect to both conception and applicability: The time--dependent
Hartree--Fock method solves the equation of motion for a Hermitian density
operator as initial value problem, with the colliding fragments in a continuum
state of relative motion. With no specification of the final state, the method
is restricted to inclusive reactions. The time--dependent mean field method, as
developed by Kerman, Levit and Negele as well as by Reinhardt, calculates the
density for specific transitions and thus applies to exclusive reactions. It
uses the Hubbard--Stratonovich transformation to express the full
time--development operator with two--body interactions as functional integral
over one--body densities. In stationary phase approximation and with Slater
determinants as initial and final states, it defines non--Hermitian,
time--dependent mean field equations to be solved self--consistently as
boundary value problem in time. The time--independent mean field method of
Giraud and Nagarajan is based on a Schwinger--type variational principle for
the resolvent. It leads to a set of inhomogeneous, non--Hermitian equations of
Hartree--Fock type to be solved for given total energy. All information about
initial and final channels is contained in the inhomogeneities, hence the
method is designed for exclusive reactions. A direct link is established
between the time--dependent and time--independent versions. Their relation is
non--trivial due to the non--linear nature of mean field methods.Comment: 21 pages, to be published in European Physical Journal
A Bayesian Approach to Inverse Quantum Statistics
A nonparametric Bayesian approach is developed to determine quantum
potentials from empirical data for quantum systems at finite temperature. The
approach combines the likelihood model of quantum mechanics with a priori
information over potentials implemented in form of stochastic processes. Its
specific advantages are the possibilities to deal with heterogeneous data and
to express a priori information explicitly, i.e., directly in terms of the
potential of interest. A numerical solution in maximum a posteriori
approximation was feasible for one--dimensional problems. Using correct a
priori information turned out to be essential.Comment: 4 pages, 6 figures, revte
Continuum Singularities of a Mean Field Theory of Collisions
Consider a complex energy for a -particle Hamiltonian and let
be any wave packet accounting for any channel flux. The time independent
mean field (TIMF) approximation of the inhomogeneous, linear equation
consists in replacing by a product or Slater
determinant of single particle states This results, under the
Schwinger variational principle, into self consistent TIMF equations
in single particle space. The method is a
generalization of the Hartree-Fock (HF) replacement of the -body homogeneous
linear equation by single particle HF diagonalizations
We show how, despite strong nonlinearities in this mean
field method, threshold singularities of the {\it inhomogeneous} TIMF equations
are linked to solutions of the {\it homogeneous} HF equations.Comment: 21 pages, 14 figure
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