From Heisenberg matrix mechanics to semiclassical quantization: Theory and first applications
- Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396 (United States)
- Department of Physics, National Taiwan University, Tapai, Taiwan 10764, Republic of (China)
Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a different and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several modified applications of EBK quantization. {copyright} {ital 1996 The American Physical Society.}
- OSTI ID:
- 383520
- Journal Information:
- Physical Review A, Journal Name: Physical Review A Journal Issue: 3 Vol. 54; ISSN 1050-2947; ISSN PLRAAN
- Country of Publication:
- United States
- Language:
- English
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