Inverse quantum mechanics

Up to now we have formulated the learning problem
in terms of a function
having a simple, e.g., pointwise, relation to .
Nonlocalities in the relation between and
was only due to the
normalization condition,
or, working with the distribution function,
due to an integration.
*Inverse problems for quantum mechanical systems*
provide examples
of more complicated, nonlocal
relations between likelihoods
=
and the hidden variables
the theory is formulated in.
To show the flexibility of Bayesian Field Theory
we will give in the following a short introduction
to its application to inverse quantum mechanics.
A more detailed discussion of inverse quantum problems
including numerical applications
can be found in
[133,143,142,138,222].

The state of a quantum mechanical systems can be completely described by giving its density operator . The density operator of a specific system depends on its preparation and its Hamiltonian, governing the time evolution of the system. The inverse problem of quantum mechanics consists in the reconstruction of from observational data. Typically, one studies systems with identical preparation but differing Hamiltonians. Consider for example Hamiltonians of the form , consisting of a kinetic energy part and a potential . Assuming the kinetic energy to be fixed, the inverse problem is that of reconstructing the potential from measurements. A local potential = is specified by a function . Thus, for reconstructing a local potential it is the function which determines the likelihood = = = and it is natural to formulate the prior in terms of the function = . The possibilities of implementing prior information for are similar to those we discuss in this paper for general density estimation problems. It is the likelihood model where inverse quantum mechanics differs from general density estimation.

Measuring quantum systems
the variable corresponds
to a hermitian operator .
The possible outcomes of measurements are given by
the eigenvalues of ,
i.e.,

(338) |

(339) |

In the simplest case, where the system
is in a pure state, say the ground state
of
fulfilling

(342) |

(343) |

In contrast to ideal measurements on classical systems, quantum measurements change the state of the system. Thus, in case one is interested in repeated measurements for the same , that density operator has to be prepared before each measurement. For a stationary state at finite temperature, for example, this can be achieved by waiting until the system is again in thermal equilibrium.

For a Maximum A Posteriori Approximation
the functional derivative of
the likelihood is needed.
Thus, for reconstructing a local potential
we have to calculate

(344) |

For that purpose, we take the functional derivative of Eq. (341), which yields

(346) |

(347) |

(348) |

The MAP equations for inverse quantum mechanics are obtained by including the functional derivatives of the prior term for . In particular, for a Gaussian prior with mean and inverse covariance , acting in the space of potential functions , its negative logarithm, i.e., its prior error functional, reads

(350) |

(351) |

The Bayesian approach to inverse quantum problems is quite flexible and can be used for many different learning scenarios and quantum systems. By adapting Eq. (349), it can deal with measurements of different observables, for example, coordinates, momenta, energies, and with other density operators, describing, for example, time-dependent states or systems at finite temperature [143].

The treatment of bound state or scattering problems for quantum many-body systems requires additional approximations. Common are, for example, mean field methods, for bound state problems [55,197,27] as well as for scattering theory [78,27,140,141,130,131,223]. Referring to such mean field methods inverse quantum problems can also be treated for many-body systems [142].