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Table 82 Micro nance Institution Activity (as of December 31, 2006) Number of MFIs Reporting (1997 2006) Number of MFIs Reporting in 2007 Only Percent Poorest Clients Represented by 873 MFIs Reporting in 2007 Total Number of Clients (as of 12/31/06) Total Number of Poorest Clients (as of 12/31/06) Total Number of Poorest Women (as of 12/31/06) Number of Poorest Family Members Affected (as of 12/31/06)

This equality generalizes (625) and can also be viewed as a de nition of the sigmaspin coherent states It is actually the Perelomov genuine de nition

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We now proceed with the presentation of a rst (small, but instructive) panorama of applications of the standard coherent state and spin coherent state to some problems encountered in physics, quantum physics, statistical physics, and so on The selected models that are illustrated as examples in the present chapter were chosen because of their high pedagogical and illustrative content Coherent States and the Driven Oscillator The driven oscillator is a pedagogical model for presenting the S matrix as a unitary operator transforming an initial state into a nal state By introducing coherent states into the formalism as was done by Carruthers and Nieto [79], we show this operator to be nothing other than the displacement operator D (z) Also, there will appear some interesting discrete probability distributions that generalize the Poisson distribution and for which there exists a limpid physical interpretation A Nice Application of Standard or Spin Coherent States in Statistical Physics: Superradiance This second part of this chapter is devoted to a nice example of the application of the coherent state formalism The object pertains to atomic physics: two-level atoms in resonant interaction with a radiation eld (Dicke model and superradiance) The content is based on two selected ancient papers, the rst one by Wang and Hioe [80], and the second one, more mathematically oriented, by Hepp and Lieb [81] Application to Quantum Magnetism Inspired by the Perelomov monograph [10], we explain how the spin coherent states can be used to solve exactly or approximately the Schr dinger equation for certain systems, such as a spin interacting with a variable magnetic eld Again, there will appear discrete probability distributions involving spin spherical harmonics and that generalize the binomial distribution Classical and Thermodynamical Limits Coherent states are useful in thermodynamics Following a paper by Lieb [19], we establish a representation of the partition function for systems of quantum spins in terms of coherent states After introducing the so-called Berezin Lieb inequalities, we show how that coherent.





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Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5

The Newton equation for the classical version of the driven oscillators is given by m + kx = F (t) , x (71)

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where F (t) represents the time-dependent driving force Hence, we have to add to 1 the free oscillator energy 2m p 2 + 1 kx 2 the interaction potential energy V int (x, t) = 2 x F (t) The Hamiltonian of the quantum version of this model is just obtained by replacing x by Q = l c a + a , l c = 1/ 2 m and p by P = i p c a a , m /2, while we still consider the driving force as classical: pc = H= 1 2 1 2 1 P + kQ Q F (t) = aa + 2m 2 2 l c (a + a ) F (t) (72)

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In a rst approach, let us suppose that there exists just a constant driving: F (t) = F 0 By shifting the lowering operator to b =a lcF 0 Id , [b, b ] = I d , (73)

1 ( p n p n )2 4 2

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This is an in nite antisymmetric real matrix The respective spectra of the nite matrix approximations of this operator and of position and momentum operators are compared in Figures 151 and 152 for various values of the regulator = /L = in units = 1 and L = When assumes large values, the eigenvalues of [Q, P ] accumulate around i, that is, they become almost canonical

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