FROM HEISENBERG MATRIX-MECHANICS TO SEMICLASSICAL QUANTIZATION - THEORY AND FIRST APPLICATIONS

Citation
Wr. Greenberg et al., FROM HEISENBERG MATRIX-MECHANICS TO SEMICLASSICAL QUANTIZATION - THEORY AND FIRST APPLICATIONS, Physical review. A, 54(3), 1996, pp. 1820-1837
Citations number
50
Categorie Soggetti
Physics
Journal title
ISSN journal
10502947
Volume
54
Issue
3
Year of publication
1996
Pages
1820 - 1837
Database
ISI
SICI code
1050-2947(1996)54:3<1820:FHMTSQ>2.0.ZU;2-T
Abstract
Despite the seminal connection between classical multiply periodic mot ion and Heisenberg matrix mechanics and the massive amount of work don e on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, neverthel ess, a number of previously unexploited aspects of this relationship t hat bear on the quantum-classical correspondence. In particular, we em phasize a quantum variational principle that implies the classical var iational principle for invariant tori. We also expose the more indirec t 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 bo und states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heis enberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semicla ssical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest sever al modified applications of EBK quantization.