Papers on Bayesian Quantum Theory

[Joerg_Lemm]

Dear Colleagues,

The following papers are available at 
              http://pauli.uni-muenster.de/~lemm/

1. Bayesian Reconstruction of Approximately Periodic Potentials 
   for Quantum Systems at Finite Temperatures. (Lemm, Uhlig, Weiguny)
2. Inverse Time--Dependent Quantum Mechanics. (Lemm)
   (to appear in Phys. Lett. A)
3. Bayesian Inverse Quantum Theory. (Lemm, Uhlig)
   (to appear in Few-Body Systems)
4. Hartree-Fock Approximation for Inverse Many-Body Problems. (Lemm,Uhlig)
   Phys. Rev. Lett. 84, 4517--4120 (2000)
5. A Bayesian Approach to Inverse Quantum Statistics. (Lemm,Uhlig,Weiguny)
   Phys. Rev. Lett. 84, 2068-2071 (2000)

In this series of papers a nonparametric Bayesian approach 
is developed and applied to the inverse quantum problem
of reconstructing potentials from observational data.
While the specific likelihood model of quantum mechanics
may be mainly of interest for physicists, the presented techniques 
for constructing adapted situation specific prior processes 
(Gaussian processes for approximate invariances, mixtures of 
Gaussian processes, hyperparameters and hyperfields)
are also useful for general empirical learning problems
including density estimation, classification and regression.


PAPERS:

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Bayesian Reconstruction of Approximately Periodic Potentials 
for Quantum Systems at Finite Temperatures.

 by Lemm, J. C. , Uhlig, J., and A. Weiguny 

 MS-TP1-00-4, arXiv:quant-ph/0005122

http://pauli.uni-muenster.de/~lemm/papers/pp.ps.gz

Abstract:
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. Such potentials might for example correspond to periodic surfaces
modified by point defects and observed by atomic force microscopy. 
The feasibility of the approach is demonstrated for a numerical model.

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Inverse Time--Dependent Quantum Mechanics.

 by Lemm, J. C.

 MS-TP1-00-1, arXiv:quant-ph/0002010 (to appear in Phys.Lett. A)

http://pauli.uni-muenster.de/~lemm/papers/tdq.ps.gz

Abstract:
Using a new Bayesian method for solving inverse quantum problems,
potentials of quantum systems are reconstructed from time series 
obtained by coordinate measurements in non--stationary states. 
The approach is based on two basic inputs: 
1. a likelihood model, providing the probabilistic description 
of the measurement process as given by the axioms of quantum mechanics, and 
2. additional "a priori" information
implemented in form of stochastic processes over potentials.

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Bayesian Inverse Quantum Theory.

 by Lemm, J. C. and Uhlig, J.

 MS-TP1-99-15, arXiv:quant-ph/0006027 (to appear in Few-Body Systems)

http://pauli.uni-muenster.de/~lemm/papers/biqt.ps.gz

Abstract:
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.

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Hartree-Fock Approximation for Inverse Many-Body Problems.

 by Lemm, J. C. and Uhlig, J.
                  
 MS-TP1-99-10, arXiv:nucl-th/9908056
 Phys. Rev. Lett. 84, 4517--4120 (2000)

http://pauli.uni-muenster.de/~lemm/papers/ihf3.ps.gz

Abstract:
A new method is presented to reconstruct the potential of a 
quantum mechanical many--body system from observational data,
combining a nonparametric Bayesian approach with a Hartree--Fock approximation.
"A priori" information is implemented as a stochastic process, 
defined on the space of potentials.The method is computationally feasible
and provides a general framework to treat inverse problems 
for quantum mechanical many--body systems.

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A Bayesian Approach to Inverse Quantum Statistics.

 by Lemm, J. C., Uhlig, J., and  Weiguny, A.

 MS-TP1-99-6,  arXiv:cond-mat/9907013
 Phys. Rev. Lett. 84, 2068-2071 (2000)

http://pauli.uni-muenster.de/~lemm/papers/iqs.ps.gz

Abstract:
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 on 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
in terms of the potential of interest. A numerical solution in 
maximum a posteriori approximation is obtained for one--dimensional problems.
The number of measurements being small compared to the degrees of freedom
of a nonparametric estimate, the results depend strongly on the implemented 
a priori information.


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Dr. Joerg Lemm                      
Universitaet Muenster                        Email: lemm at uni-muenster.de     
Institut fuer Theoretische Physik            Phone:     +49(251)83-34922
Wilhelm-Klemm-Str.9                            Fax:     +49(251)83-36328
D-48149 Muenster, Germany             http://pauli.uni-muenster.de/~lemm      
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