Helicity quantum number

The linewidth for the 0 = 1 states shows quite the opposite dependence on J. It first rises with J and reaches a maximum around J = 3. In addition, it is significantly smaller than the linewidths for the 0=0 resonances. Unlike the latter, the 0=1 quasi-bound states cannot directly decay to the j = 0 asymptotic channel which has the helicity quantum number 0 = 0. Transitions between different helicity states can be promoted only through Coriolis coupling, i.e., the last two terms on the right-hand side of (11.7). Coriolis coupling, on the other hand, is diagonal in the rotational quantum number j. Therefore, the j = 1,0 = 1 resonances can only decay via a two-step process of the form... 

QM calculations have been performed by J.F. Castillo [28] on the exeited l A" DK PES following a different hyperspherical coordinate scheme described in detail elsewhere [31]. In this case, the convergence of the calculations is less costly than on the ground state PES and only reejuires helicity quantum numbers K = 0 — 3. The number of coupled channels is small and constant c(iual to 294 for calculations at total angular momentum J > 3. Only partial waves with J < 17 were needed to obtain converged results. 

The important point to note is that there are three summations. These are over the helicity quantum numbers A, and over the total angular momentum quantum number J. 

In contrast, the Coriolis coupling [i.e., the last two entries on the right-hand side of Equation (11.7)] couples states with different helic-ity quantum numbers fl but it is diagonal in the rotational quantum number j. 

Figure 23 Calculated resonance widths F as function of the rotational quantum number J for vibrational states (7,0,0) and (8,0,0) of HOCl. The helicity is K = 0. The two dashed horizontal lines indicate the range of the measured widths. Ref. 265. Reprinted, with permission of the American Institute of Physics, from Ref. 69.

Thus, the lowest gluonic intermediate state which has the right quantum numbers to couple to a 1 particle has three gluons, just as ortho-positronium decays into at least three real photons. The J/ decay is now visualized as proceeding via the quark-gluon diagreuns of Fig. 11.6, where quarks are denoted by continuous lines and gluons by helices. 

Fig. 6. Integral cross section versus helicity projection quantum number m for Eg = 0.3 eV.

E = V x C. This implies an interesting interpretation of the Hopf index n, since that helicity is equal to the classical expression of the difference between the numbers of right-handed and left-handed photons contained in the field Nr — Nr (defined by substituting Fourier transform functions for creation and annihilation operators in the quantum expression). In other words, n = Nr — Nr- This establishes a relation between the wave and the particle understanding of the idea of helicity, that is, between the curling of the force lines to one another and the difference between right- and left-handed photons contained in the field. 

This is what we were looking for. In quantum electrodynamics, the right-hand side of (60) is interpreted as the helicity operator, that is, the difference between the numbers of right-handed and left-handed photons. We can write the usual... 

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