Semiclassical quantization of nonseparable systems

Abstract

The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.

From Heisenberg matrix mechanics to semiclassical quantization: Theory and first applications

“…Though a version of this principle was suggested more than three decades ago [36], and subsequently several publications have been devoted to its exposition and further development [37][38][39], it appears to be largely unknown by the community at large. Except for one brief allusion, [21], it has not been applied to the problem at hand.…”

“…Before providing any further details, we must mention the problem of labeling of the eigenvalues and eigenstates of H. In our earlier work [21], we "naturally" assumed that the labeling could be done by a choice of N integers n = (n 1 , ..., n N ) of which we could keep track as we tuned one or more coupling parameters, starting from values for which the problem was integrable. The same assumption was discussed rather more thoroughly by Percival [14][15][16] who emphasized that the validity of this assumption is coterminous, in the semiclassical limit, with the existence of invariant tori.…”

“…VII. The action I i is given by the equivalent expressions [13,17,21], 8) where, in the last form, p is the momentum vector of the system. This completes the formal statement of the semiclassical quantization for invariant tori.…”

“…Though some of the work cited is based, in the sense of the correspondence principle, on this passage between quantum and classical descriptions, and the passage itself is in some respects well known (see [6][7][8], for instance), the main thesis of this paper is that this route for passing from the quantum to classical theory has not been fully explored. A possible explanation is that although the basic correspondence is well-known, it is hardly known outside the circle of the authors and former associates that matrix mechanics can be derived from a novel variational principle [36][37][38][39] called the trace variational principle (see also [40]) and that the classical limit of this principle is the variational principle for invariant tori [21].…”

“…Equations of motion for the Fourier coefficients are obtained in either Hamiltonian or Lagrangian form, and for each there is an associated variational principle. The equations of motion are solved either perturbatively or non-perturbatively and the quantized actions and energy calculated in terms of the known Fourier coefficients [13][14][15][16][17][18][19][20][21]. This is the approach of paramount interest to us.…”