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.