M-indeterminate distributions in quantum mechanics and the non-overlapping wave function paradox

Highlights: • The relative phase of non-overlapping wave functions cannot be determined by its moments. • The relative phase can be determined by other expectation values. • Non-overlapping wave functions yield many M-indeterminate distributions. - Abstract: We consider the non-overlapping wave function paradox of Aharanov et al., wherein the relative phase between two wave functions cannot be measured by the moments of position or momentum. We show that there is an unlimited number of other expectation values that depend on the phase. We further show that the Wigner distribution is M-indeterminate, that is, a distribution whose moments do not uniquely determine the distribution. We generalize to more than two non-overlapping functions. We consider arbitrary representations and show there is an unlimited number of M-indeterminate distributions. The dual case of non-overlapping momentum functions is also considered.