Phase factor

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

Defining EJh + oij, replacing v /(-co) by v r(0), since the difference is only a phase factor, which exactly cancels in the bra and ket, and assuming that the electric field vector is time independent, we find... 

Phase factors of this type are employed, for example, by the Baer group [25,26]. While Eq. (34) is strictly applicable only in the immediate vicinity of the conical intersection, the continuity of the non-adiabatic coupling, discussed in Section HI, suggests that the integrated value of (x Vq x+) is independent of the size or shape of the encircling loop, provided that no other conical intersection is encountered. The mathematical assumption is that there exists some phase function, vl/(2), such that... 

The sum excludes m = n, because the derivation involves the vector product of (n Vq H n) with itself, which vanishes. The advantage of Eq. (43) over Eq. (31) is that the numerator is independent of arbiriary phase factors in n) or m) neither need be single valued. On the other hand, Eq. (43) is inapplicable, for the reasons given above if the degenerate point lies on the surface 5. 

As mentioned in the introduction, the simplest way of approximately accounting for the geomehic or topological effects of a conical intersection incorporates a phase factor in the nuclear wave function. In this section, we shall consider some specific situations where this approach is used and furthermore give the vector potential that can be derived from the phase factor. 

In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

Figure 2. Quantum classical cross-sections for the reaction D-I-Ha (r — l,j — 1) DH (v — l,/)-l-H at 1.8-eV total energy as a function of /. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with geometric phase effect included using either a complex phase factor (dashed) or a vector potential (dotted).

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

Longuet-Higgins corrected the multivaluedness of the elechonic eigenfunctions by multiplying them with a phase factor, namely,... 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

We also describe a tracing method to obtain the phases after a full cycling. We shall further consider wave functions whose phases at the completion of cycling differ by integer multiples of 2jc (a situation that will be written, for brevity, as 2Nn ). Some time ago, these wave functions were shown to be completely equivalent, since only the phase factor (viz., is observable... 

Several years ago Baer proposed the use of a mahix A, that hansforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial diffei ential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

The first and second terras contain phase factors identical to those previously met in Eq. (82). The last term has the new phase factor [Though the power of q in the second term is different from that in Eq. (82), this term enters with a physics-based coefficient that is independent of k in Eq. (82), and can be taken for the present illustration as zero. The full expression is shown in Eq. (86) and the implications of higher powers of q are discussed thereafter.] Then a new off-diagonal matrix element enlarged with the third temi only, multiplied by a (new) coefficient X, is... 

Proof When the time-deiiendent Schiodinger equation is solved under adiabatic conditions, the upper, positive energy component has the coefficient the dynamic phase factor x C, where... 

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