31 research outputs found
Some remarks on the determination of quantum states by measurements.
The problem of state determination of quantum systems by the probability distributions of some observables is considered. In particular, we review a question already asked by W. Pauli, namely, the determination of pure states of spinless particles by the distributions of position and momentum. In this context we give a new example of two wave functions differing by a piecewise constant phase having the same position and momentum distributions. ThePauli problem is investigated also under incorporation of special types of the Hamiltonian. Moreover, in case of spin-1 systems with three-dimensional Hilbert space, it is shown that the probabilities for the values of six suitably chosen spin components determine their state
The structure of classical extensions of quantum probability theory
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed
Moment operators of the Cartesian margins of the phase space observables
The theory of operator integrals is used to determine the moment operators of
the Cartesian margins of the phase space observables generated by the mixtures
of the number states. The moments of the -margin are polynomials of the
position operator and those of the -margin are polynomials of the momentum
operator.Comment: 14 page
Semispectral measures as convolutions and their moment operators
The moment operators of a semispectral measure having the structure of the
convolution of a positive measure and a semispectral measure are studied, with
paying attention to the natural domains of these unbounded operators. The
results are then applied to conveniently determine the moment operators of the
Cartesian margins of the phase space observables.Comment: 7 page
Quantum particles from coarse grained classical probabilities in phase space
Quantum particles can be obtained from a classical probability distribution
in phase space by a suitable coarse graining, whereby simultaneous classical
information about position and momentum can be lost. For a suitable time
evolution of the classical probabilities and choice of observables all features
of a quantum particle in a potential follow from classical statistics. This
includes interference, tunneling and the uncertainty relation.Comment: 19 page
On the coexistence of position and momentum observables
We investigate the problem of coexistence of position and momentum
observables. We characterize those pairs of position and momentum observables
which have a joint observable
Optimal measurements in quantum mechanics
Four common optimality criteria for measurements are formulated using
relations in the set of observables, and their connections are clarified. As
case studies, 1-0 observables, localization observables, and photon counting
observables are considered.Comment: minor correction