Read e-book online Case Studies in Atomic Physics PDF

By E. McDaniel (Eds.)

ISBN-10: 0444108823

ISBN-13: 9780444108821

ISBN-10: 0720403316

ISBN-13: 9780720403312

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And, further because o f the form o f the relation, complete correspondence identities will always be possible for these systems through the spectral operator kernel (and sometimes, as well, through the propa­ gator and Green functions). T o illustrate the points we have just made we consider now those systems, other than ones with quadratic Lagrangians, for which complete correspondence iden­ tities are known to exist. Little more needs to be said about the bound states o f the Coulomb potential except that the existence o f the complete correspondence identity can now be regarded as a consequence o f 0 ( 4 ) symmetry.

What is important about this result is that it may be used to show that the spec­ tral operator kernel o f the/-dimensional hydrogen atom (w hen / is od d) can be ex­ pressed in terms o f the 3-dimensional kernel. It thus follows therefore from what we have shown earlier that the spectral operator kernel o f the /-dimensional hydro­ gen atom must also be expressible in terms o f the spectral operator o f a particle on a ring (w h e n / is odd). Further, bacause a complete correspondence identity exists for the particle on a ring via the spectral operator, then a complete correspondence identity must also occur for the/-dimensional bound-state Coulomb problem through the spectral operator.

Nowhere, however, have we really explained why we should expect a complete correspondence identity to exist for the Coulomb potential. It is true that Gutzwiller’ s work [50] did lead us to ex­ pect a complete correspondence identity for the spectral operator, but w e should note that Gutzwiller had no a p riori reason for suspecting the phase-integral approx­ imation for the momentum-space Green functions to be exact at the bound-state energy levels; neither does he offer any explanation o f this fact.

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Case Studies in Atomic Physics by E. McDaniel (Eds.)

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