We have studied the inverse problem of reconstructing a quantum mechanical potential from empirical measurements. The approach presented in this paper is based on Bayesian statistics which has already been applied successfully to many empirical learning problems. For quantum mechanical systems, empirical data enter the formalism through the likelihood function as defined by the axioms of quantum mechanics. Additional a priori information is implemented in form of stochastic processes. The reconstructed potential is then found by maximizing the Bayesian posterior density.
The specific advantage of this new, nonparametric Bayesian approach to inverse quantum theory is the possibility to combine heterogeneous data, resulting from arbitrary quantum mechanical measurements, with a flexible and explicit implementation of a priori information.
Two numerical examples -- the reconstruction of an approximately periodic potential and of a strictly symmetric potential -- have demonstrated the computational feasibility of the Bayesian approach for one-dimensional systems. While a direct numerical solution is thus possible for one-dimensional problems, it becomes computationally demanding for two- or three dimensional problems.
As a possible approximation scheme for many-body systems an inverse Hartree-Fock approach has been proposed. An implementation of a corresponding reconstruction algorithm has been tested for a system of fermions, for which we were able to solve the inverse Hartree-Fock equation numerically.
Finally, we want to emphasize the flexibility of the Bayesian approach which can be easily adapted to a variety of different empirical learning situations. This includes, as we have seen, inverse problems in quantum theory at zero and at finite temperature, for single particles as well as for few- or many-body systems.