Ching-Teh Li

Semiclassical quantization of nonseparable systems

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.

An equations-of-motion method for anharmonic vibrations: Application to a degenerate shell model

Abstract An equations-of-motion method is proposed for calculating anharmonic effects in doubly even spherical nuclei. The method can be viewed as an extension of the QRPA theory for spherical nuclei, and its applicability is demonstrated for the case of a degenerate shell model of configuration ( 21 2 ) 6 by comparison with the results of a previous exact diagonalization.

Variational principles and Heisenberg matrix mechanics

If in Heisenbergs equations of motion for a problem in quantum mechanics (or quantum field theory) one studies matrix elements in the energy representation and by use of completeness conditions expresses the equations solely in terms of matrix elements of the canonical variables, and if one does likewise with the associated kinematical constraints (commutation relations), one arrives at a formulation - largely unexplored hitherto - which can be exploited for both practical and theoretical development. In this contribution, the above theme is developed within the framework of one-dimensional problems. It is shown how this formulation, both dynamics and kinematics, can be derived from a new variational principle, indeed from an entire class of such principles. A powerful method of diagonalizing the Hamiltonian by means of computations utilizing these equations is described. The variational method is shown to be particularly useful for the study of the regime of large quantum numbers. The usual WKB approximation is seen to be contained as well as a basic for the study of systematic corrections to it. Further applications in progress are mentioned.

A nonlinear extension of the RPA and its application to the even nickel isotopes

Abstract A nonlinear extension of the RPA is proposed and applied to calculate anharmonic effects in the even nickel isotopes. It is shown, by comparison with an exact diagonalization, to be highly accurate.

Application of Hamilton’s principle to the study of the anharmonic oscillator in classical mechanics

A form of Hamilton’s principle for classical mechanics, appropriate to the study of arbitrary self‐sustained vibrations in one dimension is presented. It is applied as an approximate computational tool to the study of several examples of anharmonic oscillation. The trial function is a finite Fourier series chosen to approach the exact solution as the number of terms increases without limit. Analytic approximations can be obtained for the limits of weak and strong anharmonicity. Numerical results for the amplitude‐dependent frequencies compare favorably with the exact solutions.