Determination of the Quantum State by Measurements

A question raised by Pauli as to whether or not the probability densities in space and momentum determine the wave function is answered negatively. The assertion by Kemble that the probability density and its time derivative determine the wave function is shown to be not generally true. It is shown that measurement of the probability density and the probability current determine the wave function of a spinless particle. The measurement of the probability current is discussed. Measurements which determine the spin state and the density matrix of a mixture are also considered.