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

By E. McDaniel (Eds.)

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Given that a representation exists in which the action functions o f the system o f interest become identical in form to those o f some other system for which a complete correspondence identity is known to exist, and 50 A . Norcliffe, Correspondence identities and the C ou lo m b potential 2. Given also that it is possible to express the corresponding operator kernel o f the system o f interest in terms o f that o f the other system (through which the complete correspondence identity is provided) in such a way that preserves its clas­ sical path form (as does eq.

1 3 ) can be regarded as a sum over classical } 48 A , Norcliffe, Correspondence identities and the C ou lom b potential paths o f terms involving the classical action functions o f the electron in the H-atom. In other words, from eq. 11), it follows that J v(u , Uq) is given exactly as a sum over classical paths. The only problem now, then, is that o f explaining w h y I e ( 6 , 9 q) is given as a sum over classical paths in the first place. This may be done by evaluating the kernel directly by path integration showing that only the classi­ cal paths contribute.

Solutions o f the Schrödinger equation. Classical trajectories and quantities evaluated along them, such as the action, play crucial roles in the theories, and for this reason it is impor­ tant to understand exactly what the mechanisms are in terms o f classical paths that give rise to the specific quantal effects mentioned above. Here the study o f the correspondence identities and the way they have been explained, is particularly useful. The complete correspondence identities o f eqs. 4 7 ) provide us with both classical as well as quantal descriptions o f phenomena such as discrete energy levels and barrier penetration which occur in the Coulomb problem.

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