Exchange symmetry

Strangely enough, the universe appears to be eomprised of only two kinds of paitieles, bosons and fermions. Bosons are symmetrical under exehange, and fermions are antisymmetrieal under exehange. This bit of abstiaet physies relates to our quantum moleeular problems beeause eleetions are femiions. 

By Max Bom s postulate, the produet of /(a ) and its complex conjugate r / (A ) times an infinitesimal volume element d x is proportional to the probability that a paitiele will be in the volume element d x 

For the probable loeation of two paitieles to be identieal under the exchange operation, x X2 X2X, the wave funetion before exehange must be exaetly equal to the wave funetion after exehange times a phase faetor c  

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles 

There are only two ways this can be true. Either the phase factor is 1 or it is —1, that 

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Until very recently, however, the same could not be said for reactive systems, which we define to be systems in which the nuclear wave function satisfies scattering boundary conditions. It was understood that, as in a bound system, the nuclear wave function of a reactive system must encircle the Cl if nontrivial GP effects are to appear in any observables [6]. Mead showed how to predict such effects in the special case that the encirclement is produced by the requirements of particle-exchange symmetry [14]. However, little was known about the effect of the GP when the encirclement is produced by reaction paths that loop around the CL... 

So far, we have treated the atoms as distinguishable particles, both in the general theory of Section II and in the application to H + H2 in Section III. Here, we explain how to incorporate the effects of particle exchange symmetry. First, we discuss how the symmetry of the system maps from the physical onto the double space, and then explain what effect the GP has on wave functions of reactions that (like H + H2) have identical reagents and products. 

Symmetry. For the dissimilar rare gas pairs, the subject matter of this Section, exchange symmetry of the atoms does not exist. We will return to a discussion of symmetry in the next Chapter, when discussing binary induced spectra of like molecular pairs. 

The second transformation is again Schofield s, and the first one has been added to assure the exact reproduction of the quantum mechanical ideal gas behavior if particles exchange symmetry effects may be neglected [318, 286]. 

About exchange symmetry. We compute the symmetry parameter ys appearing in Eq. 6.22 for given para-H2 and ortho-H2 densities, np, n0. We assume low temperatures so that all para-H2 is in the j = 0, and all ortho-H2 in the j — 1 state. (Symmetry should not matter at temperatures above roughly 20 K, unless spectral dimer features are considered.)... 

For the treatment of a mixture of para-H2 and ortho-H2, we next compute reven> the fraction of H2 pairs associated with even (, and r0dd> the fraction associated with odd f, and rno, the fraction without exchange symmetry (para-H2-ortho-H2 pairs). Particle conservation requires... 

The computation of the fractions r is straightforward. For equilibrium, the para-H2 and ortho-H2 densities, np, n0, are obtained from the partition sum of rotational states, but if an excess of para-H2 is to be accounted for (as is often the case in astrophysics), then the np, n0 are input in some form. The number of para-H2 pairs in the volume V is n2V2 which enter the expression for reven with a multiplicity of unity. The number of ortho-H2 pairs is jn2V2 of which the fraction 5/9 enters reven, and the fraction 4/9 enters r0dd- The number of para-H2-ortho-H2 pairs without exchange symmetry is npn0V2, and the total number of H2-H2 pairs equals (np + n0)2V2. Hence, we have... 

For moment calculations, the accounting for the spectral moments hardly affects the spectral moments at the temperatures considered the extreme variation of the symmetry parameter ys from 0 to 1 modifies the moments M by less than 1% at 40 K, and much less at higher temperatures. It has been previously reported [315] that exchange symmetry matters at temperatures of less than 10 K. At the higher temperatures, one may often neglect symmetry (ys = 0 in Eq. 6.22) unless dimer features are considered the dimer structures of like pairs are in general dependent on symmetry at any temperature. 

As with homonuclear diatomic molecules, the letters s and a are used to give the nuclear-exchange symmetry of / ns for polyatomic molecules. 

It is interesting to consider the CILS spectra of isotopes. For example, the spectra of He pairs differ from those of He pairs. One reason is related to the fact that He is a fermion but He is a boson these require different exchange symmetries of the pair wavefunctions. While at room temperature (where the spectra have been taken) much of the differences that arise from the different... 

Low Energy Regime Particle-Exchange Symmetry Geometric Phase Effect... 

Finally, we wish to note that hydrogen is not the only small molecule, where molecular exchange symmetry causes the existence of a para- and an ortho- spin isotopomer. Water is another important example. In the gas phase it exists as para- or ortho-water. They are distinguishable by IR. Their concentration ratio is used in astronomy as a remote temperature sensor. The spin conversion mechanisms of these isotopomers are still an open field for future studies [98, 99]. 

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