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 ‖ψ〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution ‖ψ(x,t)${\mathrm{\Vert }}^2$ 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 Δt later, ‖ψ(x,t+Δt)${\mathrm{\Vert }}^2$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, ${\mathcal{T}}^{\mathit{M}}$. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. © 1996 The American Physical Society.