Observation of Moving Wave Packets Reveals Their Quantum State

We show how to infer the quantum state of a wave packet from position probability distributions measured during the packet's motion in an arbitrary potential. We assume a nonrelativistic one-dimensional or radial wave packet. Temporal Fourier transformation and spatial sampling with respect to a newly found set of functions project the density-matrix elements out of the probability distributions. The sampling functions are derivatives of products of regular and irregular wave functions. We note that the ability to infer quantum states in this way depends on the structure of the Schrödinger equation.