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Annales De L Institut Henri Poincare-physique Theorique

A characterization of coherent states from varying Planck's constant

J. C. Houard

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

For a quantum system with one degree of freedom we introduce some families of stats depending on the Planck's constant, that we call «classical states». We show that a definite class of these families allows to characterize the coherent states

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Coherent States of Parametric Oscillators in the Probability Representation of Quantum Mechanics†

Vladimir N. Chernega, Olga V. Man’ko

arXiv:2003.01556•Journal of Russian Laser Research•Jan 1, 2020•7 citations
We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom NNN on the Planck constant $\hbar $. It is shown that to use the central limit theorem under taking the limit $N\to +\infty $ one should keep the energy of the system to be constant. In the case, the resulting distribution is Gaussian if the initial distribution is a product of independent excited states of a quantum oscillator or even and odd coherent states either. Then, if one turns the Planck constant ℏ→0\hbar \to 0ℏ→0 we get δδδ-function associated with the distribution concentrated in zero with the probability equal to one.
We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom N on Planck's constant . It is shown that the use of the central limit theorem at N→+∞ in the quantum case can be employed for checking the condition of its validity in the famous Lyapunov form. In this case, the resulting distribution is Gaussian if the initial distribution is a product of either independent excited states of a quantum oscillator or even and odd coherent states. Then, at Planck's constant → 0, one gets the δ-function associated with the distribution with the maximum at zero and with the probability equal to one.
Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras (HAs), respectively (Grgin and Petersen 1974 J. Math. Phys. 15 764, Sahoo 1977 Pramana 8 545). We show that the tensor product of two quantum HAs, each characterized by a different Planck's constant (PC), is an algebra of the same type characterized by yet another PC. The algebraic structure of mixed quantum and classical systems is then analysed by taking the limit of vanishing PC in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantum–classical systems. It shows that in the interacting mixed quantum–classical description, there can be no back-reaction of the quantum system on the classical system. A natural algebraic requirement involving restriction of the tensor product of two quantum HAs to their components proves that PC is unique.