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Physical Review A

Hierarchy of local minimum solutions of Heisenberg’s uncertainty principle

David K. Hoffman, Donald Jack Kouri

Published Apr 16, 2002• 7 citations
Publisher

Abstract

Local minimum solutions to Heisenberg's uncertainty principle are studied in detail. Of particular concern is obtaining an optimal procedure to squeeze the uncertainty in one canonical operator while introducing, in a well-defined mathematical sense, the least possible increase in the uncertainty of the relevant conjugate operator. The result is a hierarchy of ``minimum uncertainty $(\ensuremath{\mu}\ensuremath{-})$ wavelets,'' which are intimately associated with the ``Hermite distributed approximating functionals'' [D. K. Hoffman and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000)]. The $\ensuremath{\mu}$ wavelets can be used with any quantum state to squeeze the uncertainty of a particular observable. Several specific examples are given and some possible applications are discussed.

DOI
10.1103/physreva.65.052106

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Local minimum solutions to Heisenberg's uncertainty principle are studied in detail. Of particular concern is obtaining an optimal procedure to squeeze the uncertainty in one canonical operator while introducing, in a well-defined mathematical sense, the least possible increase in the uncertainty of the relevant conjugate operator. The result is a hierarchy of minimum uncertainty $(\ensuremath{\mu}\ensuremath{-})$ wavelets,'' which are intimately associated with the Hermite distributed approximating functionals'' [D. K. Hoffman and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000)]. The \ensuremathμ\ensuremath{\mu}\ensuremathμ wavelets can be used with any quantum state to squeeze the uncertainty of a particular observable. Several specific examples are given and some possible applications are discussed.
We derive a new hierarchy of local minimum Heisenberg-uncertainty states by introducing a superposition of "small waves" onto some initial state. Our objective is to increase the resolution in one observable, with the least decrease in the resolution in the conjugate observable. This leads to a constrained minimization which in a well-defined sense yields the best possible way of achieving this goal. The results are relevant to many topics (e.g., quantum optics and control, Bose-Einstein condensation, path integration, etc.).
We consider additional aspects of the recently derived “minimum uncertainty” (μ) wavelets. In particular, we show that they are fundamentally related to both the harmonic oscillator eigenstates and the canonical coherent states that play a fundamental role in quantum dynamics. In addition, we derive new raising and lowering operators that apply to the μ-wavelets. Finally, we explore in some detail the senses in which the μ-wavelets form complete sets that can be used in a variety of applications in quantum dynamics.
This paper is concerned with the uncertainty principle in the context of the ane-W eyl-Heisenberg group in one and two dimensions. As the representation of this group fails to be square integrable, we explore various admissible sections of this group, and calculate the resulting uncertainty principles as well as its minimizers with respect to these sections. Previous studies have shown that these sections give rise to mixed smoothness spaces. We demonstrate that the minimizers obtained for these sections actually interpolate between Gabor and wavelets functions.
Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum. \textcopyright{} 1996 The American Physical Society.