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that ( J, ) are canonical action angle variables is what xes the amplitude behavior of the coherent states, that is, J n e i x n /(N ( J) [x n ]!) (cf (981)) In order for coherent states to interpolate well between quantum and classical mechanics, it is necessary for values of the action J > that the quantum motion > be well approximated by the classical motion In particular, for a classical system with closed, localized trajectories, a suitable wave packet should, if possible, remain coherent for a number of classical periods For the systems under study in this chapter, we have demonstrated the tendency for improved packet coherence with increasing J values within the range studied For signi cantly larger values of J, we notice that the packet coherence substantially improves Interesting results were obtained independently in a related study by Fox and Choi [138], who found a similar packet coherence for 10 or more classical periods for an in nite square well, even though they used a different amplitude prescription for their coherent states In both approaches, however, the probability distribution shows a Gaussian behavior for large values of J, and this explains the similarity of the results It would appear that allowing for generalized phase and amplitude behavior in the de nition of coherent states has led us closer to the idealized goal of a set of coherent states adapted to a chosen system and having a large number of properties in common with the associated classical system, despite being fully quantum in their characteristics



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The appropriateness of equilibrium conditions in the studies of living matter was rst explicitly articulated by Tanford, who wrote on the occasion of the centennial anniversary of the Gibbs papers on chemical equilibrium (187, 188): The time now seems appropriate to use his method of thermodynamic analysis as a basis of thinking about the organization of the living cell and, beyond that, the assembly of cells into multicellular organisms (189) Tanford suggested that understanding of the organization of the living matter can be achieved by dividing the process into two parts, biosynthesis and assembly of biosynthetic products into organized structure The latter process should be under thermodynamic control: that is, self-organized structures represent the macromolecular arrangement providing the lowest chemical potential (which, in the absence of chemical transformations, is equal to the free energy) (189).





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This chapter is devoted to another occurrence of SU (1, 1) in the construction of various popular quantum states The so-called squeezed states, which are to be described here, pertain again to quantum optics and have been raising interest in quantum optics and other elds for the last three decades Squeezed states, a name given by Hollenhorst [139], were initially introduced in quantum optics [140] for dealing with processes in which emission or absorption of two photons is involved The formalism underlying the process involves the square of raising and lowering 2 operators, a 2 and a , for each mode of the quantized electromagnetic eld In this regard, squeezed states might be viewed as a sort of two-photon coherent states Their mathematical properties had actually been investigated before [141 143] The experimental evidence for such states was provided by Slusher et al [144] Interesting applications to quantum nondemolition in view of detecting gravitational waves were envisaged as early as 1981 [145, 146] In addition, it is interesting to learn from Nieto [147] that these states were already known in 1927, 1 year after the Schr dinger states For more details on theoretical and experimental aspects of squeezed states, see [25] and the recent report by Dell Annoa et al [148] The introduction that is given here to the squeezed coherent states insists more particularly on their relations with the unitary irreducible representations of the symplectic group S p(2, R), another version of SU (1, 1) We will also stress, of course, their importance in quantum optics: they render possible reduction of the uncertainty on one of the two noncommuting observables present in the measurements of the electromagnetic eld

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