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

Abstract

Despite the seminal connection between classical multiply-periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical (EBK) quantization of bound states, we show, that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a new and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several new or modified applications of EBK quantization.

High-Performance Algorithms for Quantum Systems Evolution

“…With F being the Generator of the transformation. It is convenient to choose F as a solution of certain equation ( [4]) that guarantees the possibility of evaluation of path integral with the weight (2). In that case instead of computation of path integral we have to solve four times the equation for F , that is more convenient since it is partial differential equation of the first order ( [3]).…”

“…Several attempts were made to overcome those difficulties by describing the evolution in classical terms and solving large time problems by classical means. A lot of work was done in establishing rigorous relationship between classical and quantum tools for description of system evolution ( [2]). Although formal mathematical solution of the problem was given in 1979 ( [3]), and computational algorithm was proposed in 1986 ( [4]), only recently it was possible to realize it on large systems ( [5]).…”

Symplectic structure for general relativity and Einstein–Brillouin–Keller quantization

“…The EBK quantization method remains under investigation to date. For example, the relationship between the EBK method and the Heisenberg matrix mechanics has been discussed in reference [10] and some elucidative examples of the method have been provided in reference [11]. The EBK quantization method has also been used to analyze quantum chaos [12,13].…”

Symplectic structure for general relativity and Einstein-Brillouin-Keller quantization

“…For example, the relationship between the EBK method and the Heisenberg matrix mechanics has been discussed in Ref. [10] and some elucidative examples of the method have been provided in Ref. [11].…”