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9 Another Family of SU(1,1) Coherent States for Quantum Systems
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Fig 911 Temporal behavior of the average position for the asymmetric P schl Teller potential ( , ) = (4, 8) with n max = 10, for (a) J = 20 and (b) J = 50 Source Antoine et al [112] (reprinted with permission from [Antoine, J-P, Gazeau, JP, Monceau, P, Klauder JR, Pernon KA, Temporally stable coherent states for in nite well, J Math Phys, 42, p 2349, 2001], American Institute of Physics)
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974 Discussion
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The speci c choices we have made here for the set of coherent states are based on two additional guiding principles besides continuity and resolution of unity, as was proposed in [58] The rst of these is temporal stability , which in words asserts that the temporal evolution of any coherent state always remains a coherent state The second of these, referred to as the action identity in [58], chooses variables for the coherent state labels that have as close a connection as possible to classical action angle variables In particular, for a single degree of freedom, the label pair ( J, ) is used to identify the coherent state | J, Temporal stability means that, under the dynamics chosen, temporal evolution proceeds according to | J, + t , for some xed parameter To ensure that ( J, ) describes action angle variables, it is suf cient to require that the symplectic potential induced by the coherent states themselves is of Darboux form, or speci cally that i J, | d | J, = J d ,
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where d | J, = | J + d J, + d | J, Temporal stability is what xes the phase behavior of the coherent states, that is, the factor e i x n (cf (977)), while ensuring
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97 Physical Aspects of the Coherent States
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Fig 912 Temporal behavior of the average position in the case of the in nite square well, for a very high value of J = 106 Source Antoine et al [112] (reprinted with permission from [Antoine, J-P, Gazeau, J-P, Monceau, P, Klauder JR, Penson KA, Temporally stable coherent states for in nite well, J Math Phys, 42, p 2349, 2001], American Institute of Physics)
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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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