Phase factor time reversal

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

Apart from the phase factor — i, the transformed spinor will be recognized as the ungerade spinor of Chapter 11. The original and time-reversed states are orthogonal and therefore degenerate, and consequently... 

In general, time reversal changes the sign of the m index and multiplies by a phase factor. 

In addition, Hso is time-reversal invariant. The time-reversal operator for a single electron is the antiunitary operator (up to an arbitrary phase factor) [25]... 

The minus sign in the spin-down orbital is included to make the wavefunction odd under time reversal). The imaginary phase factor between the two contributions to each orbital results in the FON density... 

A few examples will illustrate the notation introduced in equations (18) and (19), showing why the phase factors are needed in the relationships linking the original state (p/J) with its time-reversed partner, cpyM). Suppose, for example, that the original state is a spin-up plane wave... 

Phase 1 [identified as step 1 in Eq. (7)] is complete within the mixing time and gives a compound retaining absorbance at 340 nm, but with quenched NADH and protein fluorescence. Phase 2 is a first-order process in which absorbance at 340 nm is destroyed, protein fluorescence appears, and NADH fluorescence remains quenched. If this first-order process were after the redox step [step 4 in Eq. (7) ], then the first turnover of NADH would be very fast. This is not observed. If the first-order process were the redox step itself [step 3 in Eq. (7)] then it would be slower with NADD than with NADH by a factor of 6 to 7, as with alcohol dehydrogenase (293). No appreciable isotope effect is measured. Thus the first-order phase must be identified with an isomerization of the ternary complex with NADH [step 2 in Eq. (7)] before the redox step (269,279). Siidi (293a) has also observed two phases in the reverse reaction and has deduced the on rate for pyruvate. The kinetics do not indicate the... 

Retention factor A measure of the amount of time an analyte spends in the stationary phase relative to the mobile phase Retention time The time taken for an analyte to travel from the point of injection to the point of detection within an HPLC system Reverse phase chromatography Describes the chromatographic separation in which the stationary phase is nonpolar and the mobile phase is composed of an aqueous, moderately polar liquid Robustness A measure of a method s ability to withstand small but deliberate changes in the method parameters it provides an indication of its reliability during normal usage Selectivity factor See separation factor... 

Note that in the present derivation we avoided providing an explicit form for the inverse of the time reversal operator. As a matter of fact, while space inversion is its own inverse, applying time reversal twice may give rise to an additional phase factor, which is -b 1 for systems with an even number of electrons, but — 1 for systems with an odd number of electrons. We shall demonstrate this point later in Sect. 7.6. Hence, = , or... 

Note that the action of time reversal on the spin functions precisely corresponds to the C operator and thus is represented hy (Cj). This result may be generalized, in the sense that time reversal can be represented as the product of complex conjugation, denoted as K, and a unitary operator acting on the components of a function space, which we shall denote by the unitary matrix U. We thus write = U/f. When this operator is applied twice, it must return the same state, except possibly for a phase factor, say exp(fx). Following Wigner, we now show that the two cases = 1 are in fact the only possibilities. Hence, the phase factor can be only either H-1 (time-even state) or -1 (time-odd state) [11, Chap. 26]. Taking time reversal twice, we have... 

Whenever the PNC interaction has time-reversal symmetry, as it does in the forms of interest here given in Eqs. (8) or (11), then ElpNc and Ml will have a relative phase of ir/2. This phase maximizes the circular dichroism and optical rotation shown in Eqs. (24) and (26). We have already mentioned the importance of this phase in the qualitative discussion in Section 1.3, and we noted the factor of i in the result of the simple calculation of PNC mixing in Eq. (14). 

We have now shown that a wave function and its time reverse are energy degenerate in both the relativistic and the nonrelativistic case. One way this can happen is if time reversal just produces the same function—that is, the function is invariant under time reversal, possibly with the exception of a phase factor. In this case we have... 

While these phase factors are useful for the real groups, we would like to be able to make the Hamiltonian matrix real for any group, if possible. After all, for an even number of electrons, we can always construct a real basis, as demonstrated in section 9.3. In fact, we can transfer the principles from that section directly to the case of spin-orbit Cl and construct linear combinations of determinants that are symmetric under time reversal. 

---