HOW TO DETERMINE A QUANTUM STATE BY MEASUREMENTS - THE PAULI PROBLEM FOR A PARTICLE WITH ARBITRARY POTENTIAL

The problem of reconstructing a pure quantum state \psi] from measurab le quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \( psi(x,t)\(2) has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short -time interval Delta t later, \psi(x,t+Delta t)\(2), the set of wave f unctions compatible with these distributions is given by a smooth mani fold M in Hilbert space. The manifold M is isomorphic to an M-dimensio nal torus, F-M. Finally, M additional expectation values of appropriat ely chosen nonlocal operators fix the quantum state uniquely. The meth od used here is the analog of an approach that has been applied succes sfully to the corresponding problem for a spin system.