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

TL;DR

This paper explores the non-overlapping wave function paradox in quantum mechanics, demonstrating that phase information cannot be captured by moments alone and that the Wigner distribution is M-indeterminate, with implications for quantum state representations.

Contribution

It reveals the M-indeterminacy of the Wigner distribution and extends the analysis to multiple non-overlapping wave functions and various representations.

Findings

Moments of position or momentum do not capture phase information.

The Wigner distribution is M-indeterminate, not uniquely determined by moments.

Multiple non-overlapping wave functions lead to M-indeterminate distributions.

Abstract

We consider the non-overlapping wave function paradox of Aharanov \textit{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.