A new expression for harmonic oscillator brackets

BOUDEWIJN J. VERHAAR

Jan Dobeš

An explicit formula for general harmonic oscillator brackets is derived. The formula is particularly useful and convenient in numerical computations.

Ladislav Trlifaj

B.J. Berhaar

Saulius Mickevičius, Darius Germanas, R.K. Kalinauskas

Abstract We present a new independent scheme of SO(3) group transformations suitable for the N particle system, composed of N − 1 and 1 particle subsystems, where N − 1 particles have their own intrinsic clusterization. The simple expressions for corresponding four-particle harmonic-oscillator transformation brackets are presented, as well as their simplifications for the special values of mass ratio parameters d = 0, d → ∞ and d 1 = 0, d 1 → ∞.

Nutku, Y

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.