Group coherent state

Any algebraic operator, written in terms of the boson operators a, n of Chapter 2 can be converted into a classical operator, written in terms of the variables (or p, q). We describe here again the derivation of van Roosmalen 1982. One introduces a group coherent state... 

The general theory of classical limits of algebraic models is formulated not in terms of the group coherent states of Eq. (7.17) but rather in terms of projective coherent states. The ground-state projective coherent state is... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In order to make the END formalism practical, some approximations must be made in the representation of the waveffinctions of the electrons and nuclei. The particulars are outlined in the next section. In the simple model employed in this work, we choose to represent the electronic wavefunction by a group theoretical coherent states parametrization of a single determinant (SD). The nuclear wave function is formulated in terms of a frozen Gaussian wave packet (FGWP) and the limit of a narrow width is taken. This latter approximation corresponds to the classical limit for describing the nuclei. The coherent state representation of a single determinant leads to the so-called Thouless parametrization (29). For the description of the nuclear wavefunction, a quantum description has also been worked out (26, 30), but is yet to be implemented. 

The antiphase coherence state generated is a function of the underlying coupling constant. In a IH correlation experiments the optimum evolution period is given by 1/(2 JC X, IH). However in a real sample there will be a variety of values for J( X, IR) for different IH, x spin groups so a... 

The Gaussian wave packet in this form is the original coherent state . Generalizations of this concept have been made, in particular the work of Perelomov [14] has introduced so-called group-related coherent states. Such a state is formed by the action of a Lie group operator exp acting on a reference state lO). The Zm are the, in general... 

A more general coherent state description of a Gaussian wave packet is required when we allow the width parameter to evolve in time. The corresponding Lie group is then Sp(2, K), which is isomorphic to SU(1,1) or SO(2,1). The generators of the Sp(2, R) Lie algebra are... 

Each element of the coset space G/H then corresponds to a coherent state. The decomposition of the group into cosets, taking advantage of the stability group properties, reduces the parameter space of the coherent state to a nonredundant set. In our case the stability group is SO(2) and we can write... 

A determinantal wave function expressed in this form is a coherent state. The associated Lie group is the unitary group U K) and the reference state Eq. (127) is the lowest weight state of the irreducible representation [1 0 ] of Lf(K). The stability group is U N) X U(K — N). The norm in an orthonormal basis of spin orbitals is... 

A.M. Perelomov, Coherent states for arbitrary Lie group, Commun. Math. Phys., 26 (1972) 222-236. 

A detailed analysis of electronic coherent states associated with the one-particle unitary group is presented in terms of coset generators of this group. For even number of electrons these states form non-linear manifolds of anti symmetrized geminal powers (AGPs) that are parameterized by complex coordinates. Different classes of manifolds of AGP states are associated with distinct coset decompositions and produce different restrictions of the time-dependent Schrodinger equation. 

The research group of Th.W. Hansch succeeded in 2002 to trap for the first time cold atoms in such an optical lattice. They cooled at first the atoms below the critical temperature fort BE condensation. Then the well depth of the optical lattice was increased. This transferred the coherent state of the atoms in the free BEC, where all atoms are in the same state i.e. described by the same wavefunction and are therefore not distinguishable, into the incoherent Mott state, where each atom sits on its separate location and can be therefore distinguished from the other atoms. Decreasing the well depth brings the atoms again back into the coherent BEC state. Just by changing the well depth of the optical lattice switches the atomic ensemble from a coherent into an incoherent state and back [1204]. 

Generally, the recrystaUization of S-layer protein into coherent monolayer on phospholipid films was demonstrated to depend on (1) the phase state of the hpid film, (2) the nature of the lipid head group (size, polarity, and charge), and (3) the ionic content and pH of the subphase [122,138] (Table 6). 

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