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By E. McDaniel (Eds.)

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Senior Physics 1

Translated and revised from the 1986 Russian variation.

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Sample text

Again, the actions o f each successive trajec­ tory will differ by a fixed amount equal to the action for one orbit. This amount is a function only o f the energy and due to the total destructive interference o f the classical path contributions except for certain values o f the energy, where the inter­ ference is constructive, the resulting energy spectrum is therefore discrete. While these views o f quantization and the mechanisms responsible for quantization in the semiclassical limit are accepted as far as separable systems are concerned, opin­ ions differ as to the mechanisms responsible for the quantization o f bound nonseparable systems.

It is worthwhile commenting on the form o f this complete correspondence iden­ tity. First o f all we see that all possible classical paths running from t 0 P contri­ bute to the expression for I E {p, P q ). There is an infinite number o f contributions due to the classical trajectories being closed paths. Secondly we note that each clas­ sical path contributes to the kernel with an overall phase o f ±1 due to the sin \2tic + ω I term. I f c is positive then the overall phase is + 1 ; if c is negative it is — 1.

U ti­ lizing the special properties o f the spherical harmonics, Bander and Itzykson show that corresponding projection operator kernels in different dimensions can be re­ lated to one another b y differentiation. They show for example that the projection operator kernels defined on the ( / + l)-dimensional sphere, i f / i s odd, can be o b ­ tained directly by repeated differentiation from the corresponding kernels defined over the 4-dimensional sphere. The kernel P n{u ,u ^ ) in the case o f 0 ( 4 ) symmetry, as we have shown, is ( n sin η ω ) Ι sin ω and i f this is differentiated with respect to cos ω the result (to within a multiplicative constant) according to Bander and Itzykson is the value o f the nth projection operator kernel in the case o f 0 ( 6 ) symmetry.

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