Adiabatic invariants and the problem of quasiclassical quantization of many-dimensional systems

Frank R. Johnston, Philip Pechukas

S. P. Karkach

The Laplace method is considered in application to one-dimensional two-state eigenvalue problems in the example of simple models, including that of coupled parabolic terms and radial oscillators with quadratic, linear and constant coupling (basic models of Renner, pseudo-Jahn-Teller effects etc.). In all cases, the solution can be presented either as a two-state analogue of the Bohr-Sommerfeld quasiclassical quantization rule, or in more general form. Compared with the semiclassical picture of inelastic collisions in the momentum-space representation, the contour of integration of 'trajectory equations' for the case of bound states is finite, and any desired degree of accuracy can easily be achieved. Because no convergence problems arise, the need to compute the equation possessing 'physical sense' (e.g. for adiabatic amplitudes) is circumvented, suggesting that the method would be applicable also to complicated potentials and to many-state problems. Numerical tests for the energy level positions and calculations of eigenfunctions are presented; practical applications and generalizations are outlined.

Hiroshi Kuratsuji, Shinji Iida

Leonard Parker

We show, in an elementary but apparently new way, that the ratio of the energy to the frequency of a simple harmonic oscillator is an adiabatic invariant. The adiabatic invariance is a consequence of the conservation of angular momentum in the two-dimensional system, of which the one-dimensional harmonic oscillator is the projection. We also briefly discuss the connection between the quantization of angular momentum and the quantization of the oscillator's energy.

Abraham Klein, Ching‐Teh Li

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.

E. Gozzi