  Probability densities that are not uniquely determined by their moments are
said to be "moment-indeterminate", or "M-indeterminate". Determining whether or
not a density is M-indeterminate, or how to generate an M-indeterminate
density, is a challenging problem with a long history. Quantum mechanics is
inherently probabilistic, yet the way in which probability densities are
obtained is dramatically different in comparison to standard probability
theory, involving complex wave functions and operators, among other aspects.
Nevertheless, the end results are standard probabilistic quantities, such as
expectation values, moments and probability density functions. We show that the
quantum mechanics procedure to obtain densities leads to a simple method to
generate an infinite number of M-indeterminate densities. Different
self-adjoint operators can lead to new classes of M-indeterminate densities.
Depending on the operator, the method can produce densities that are of the
Stieltjes class or new formulations that are not of the Stieltjes class. As
such, the method complements and extends existing approaches and opens up new
avenues for further development. The method applies to continuous and discrete
probability densities. A number of examples are given.
