Trajectories, semiclassical quantization

When classical trajectories are calculated for the H2O model with two O—H stretches discussed in section 4.3.2 (p. 76), local-mode type motion is found for which energy is trapped in individual O—H bonds (Lawton and Child, 1979, 1981). The trajectories are quasiperiodic and application of the EBK semiclassical quantization condition [Eq. (2.72)] results in pairs of local-mode states in which there are n quanta in one bond and m in the other or vice versa. The pair of local-mode states (n,m) and (m,n) have symmetry-related trajectories which have the same energy. The local-mode trajectory for the (5,0) state is depicted in Figure 4.6d. 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

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