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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Equivalence Classes of Minimum-Uncertainty Packets. II

David Stoler

Published Sep 15, 1971• 283 citations
Publisher

Abstract

In a previous paper we have shown that all minimum-uncertainty packets are unitarily equivalent to the coherent states and that coherence may be viewed as stationary minimality. In this note we give some additional information relating to the nature of the unitary-equivalence structure. We also give a new calculation of some matrix elements of the operator that implements the unitary equivalence which is not subject to the shortcoming inherent in the original calculation.

DOI
10.1103/physrevd.4.1925

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In a previous paper we have shown that all minimum-uncertainty packets are unitarily equivalent to the coherent states and that coherence may be viewed as stationary minimality. In this note we give some additional information relating to the nature of the unitary-equivalence structure. We also give a new calculation of some matrix elements of the operator that implements the unitary equivalence which is not subject to the shortcoming inherent in the original calculation.
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.
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.