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American Journal of Physics

Quantum mechanical squeezed state

R. Muñoz-Tapia

Published Nov 1, 1993• 16 citations
Publisher

Abstract

The squeezed states appear in different fields under different forms. Compact operator expressions are given and some of their properties reviewed with the connections between different contexts clarified.
DOI
10.1119/1.17382

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The Entry of π: The Conditions That Force Compact Closure asks not where π appears, but what conditions make π unavoidable. The paper argues that π is usually defined too late, after the circle has already appeared, and instead proposes that π enters when distinction becomes boundary-bearing, when boundary creates an inside/outside relation, and when persistence requires completed return. The central claim is that π is the compact-closure invariant of maintained distinction. A bare difference does not force π; difference may remain open, linear, or merely oppositional. π becomes necessary only when difference is constrained into persistence, persistence requires return, and return closes sufficiently to generate a compact phase condition: θ ≡ θ + 2π. The paper develops the cascade: difference → constraint → boundary → inside/outside → closed return → π → phase → spectrum → physical form. It then connects this structure to quantum mechanics by interpreting the Schrödinger equation as a dynamical expression of phase persistence, where unitary evolution preserves the state through norm-preserving rotation in complex state space. The paper also clarifies the relation between π and light, arguing that light inherits π through phase closure rather than being reducible to π as a substance For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com
From the normally ordered form of the density operator of a squeezed coherent state (SCS), we directly derive the compact expression of the SCS's photon-number distribution (PND). Besides the known oscillation characteristics, we find that the PND is a periodic function with a period of π and extremely sensitive to phase. If the squeezing is strong enough, and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values, different oscillation behaviours in PND for even and odd photon numbers appear, respectively.