Abstract
This paper considers the problem of whether mixed states can arise in the Nth order minimum uncertainty expression UN=<Ψ‖‖x‖N‖Ψ≳<Ψ‖‖p‖N‖Ψ≳. Using a method due to R. Jackiw [J. Math. Phys. 9, 339 (1968)], and finding the extremum of the appropriate functional, a set of equations is derived which must be satisfied by such a mixed state. It is easily shown that a sufficient condition that the equations have a solution is that the mixed state in fact reduces to a pure state. It does not appear possible to rigorously prove that such a reduction is necessary, although the authors present approximate calculations which make such a conclusion plausible. In the course of the approximate calculations generalized Nth order ‘‘squeezed state’’ values for UN are also obtained which are of some interest in and of themselves.
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