Berries

Berlin green, FeFe(CN)mechanism postulated for the interchange of substituents in trigonal-bipyramidal 5-co-ordinate complexes, e g. PF, and its substituted derivatives. berthoUide compound Solid phases showing a range of composition. 

It is present in rowan berries and is characteristic of the fruits of the Rosaceae. 

Berry R S, Rice S A and Ross J 1980 Physical Chemistry (New York Wiley)... 

Berry R S 1999 Phases and phase changes of small systems Theory of Atomio and Moleoular Clusters ed J Jelllnek (Berlin Springer)... 

Barrett J J and Berry M J 1979 Photoacoustic Raman spectroscopy (PARS) using cw laser sources Appl. Phys. Lett. 34 144-6... 

Berry M 1990 Anticipations of the geometric phase Physics Today 43 34... 

Northby J A 1987 Structure and binding of Lennard-Jones clusters 13< W < 147 J. Chem. Phys. 87 6166 Berry R S 1993 Potential surfaces and dynamics what clusters tell us Chem. Rev. 93 2379... 

Berry M T, Loomis R A, Gianoarlo L C and Lester M I 1991 Stimulated emission pumping of intermoleoular vibrations... 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

Figure 3. Phase tracing for circling outside the ci pair for the model in A and states in symmetry. The Berry phase (half the angle shown at the extremity of the figure) is here —2tt.

Figure 5. Phase tracing for the case of trigonal degeneracies when the circle encompasses all four ci s and the Berry phase is 271.

Clearly, the pseudoscalar term vanishes at these points so the ci character at the roots is maintained, no matter whether there are or are not A-i terms. Also, the vanishing of Ai terms will not lead to new ci s.) On the other hand, by circling over a large radius path q oo,so that all ci s are enclosed, the dominant term in Eq. (84) is the last one and the acquired Berry phase is —4(2tc)/2 = —4ti. 

To see that this phase has no relation to the number of ci s encircled (if this statement is not already obvious), we note that this last result is true no matter what the values of the coefficients k, X, and so on are provided only that the latter is nonzero. In contrast, the number of ci s depends on their values for example, for some values of the parameters the vanishing of the off-diagonal matrix elements occurs for complex values of q, and these do not represent physical ci s. The model used in [270] represents a special case, in which it was possible to derive a relation between the number of ci s and the Berry phase acquired upon circling about them. We are concerned with more general situations. For these it is not warranted, for example, to count up the total number of ci s by circling with a large radius. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

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