Wr. Greenberg et al., FROM HEISENBERG MATRIX-MECHANICS TO SEMICLASSICAL QUANTIZATION - THEORY AND FIRST APPLICATIONS, Physical review. A, 54(3), 1996, pp. 1820-1837
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.