Note on semiclassical uncertainty relations

Abstract

An important manifestation of the Uncertainty Principle, one of the cornerstones of our present understanding of Nature, is that related to semiclassical localization in phase-space. We wish here to add some notes on the subject with reference to the canonical harmonic oscillator problem, with emphasis in the concepts of semiclassical Husimi distributions, the associated Wehrl entropy, escort distributions, and Fisher's information measure.

Fluctuations, Entropic Quantifiers and Classical-Quantum Transition

“…A very important information-quantifier is the so called Fisher's information measure (FIM) [22,23]. Its phase space representation is [24] …”

“…For some applications and details of this measure see, amongst many possibilities [22,23,25]. From Equation (7) and integrating over phase space, the Fisher measure acquires the form…”

Husimi Distribution and the Fisher Information

“…The indispensable tool in this proposal is a quasi probability called Husimi distribution [1], which is frequently employed to characterize the quantum and classical behavior [2] of systems. Also, it possesses interesting applications in several areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory and Nanotechnology [3][4][5][6][7][8][9][10]. Its main properties are: 1) it is definite positive in all phase space, 2) it possesses no correct marginal properties, 3) it permits to calculate the expectation values of observables in quantum mechanics similarly to the classical case [11], and 4) it is a special type of probability that simultaneously approximate location of position and momentum in phase space.…”

Трансформационное Свойство Функции Хусими И Ее Связь С Функцией Вигнера И Симплектическими Томограммами

“…В ра-ботах [25], [26] функция Хусими использовалась для анализа свойств зацепленности в различных сложных системах. В работах [27], [28] исследовалась связь функции Хусими с соотношениями неопределенности.…”

“…поэтому экспоненциальные операторы с двойным дифференцированием по одной переменной не меняют структуру выражения (28), которая обеспечивает его по-ложительность. Действительно, эти операторы действуют только на функции Ψ, которые преобразуются в другие функции, но в результате мы опять получаем сум-му функций, умноженных на свои комплексно-сопряженные.…”