Equations of Motion and Further Developments

K. P. Michnicki

A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.

Günther Ludwig

Umberto Lombardo, G. Giuliani, Yifei Niu

Keh-Ning Huang

An equation-of-motion approach is used to develop the relativistic many-body theory of atomic transitions. The relativistic equations of motion for transition matrices are formulated with the use of techniques of quantum-field theory. To reduce the equations of motion to a tractable form which is appropriate for numerical calculations, a graphical method to resolve the complication arising from the antisymmetrization and angular-momentum coupling is employed. The relativistic equation-of-motion method allows an ab initio treatment of correlation and relativistic effects in both closed- and open-shell many-body systems. A special case of the present formulation reduces to the relativistic random-phase approximation.

Michael G Cottam, Zahra Haghshenasfard

Evgeny Levich, Victor Yakhot

The microscopic theory of the time evolution of an interacting Bose gas being cooled to the temperature T<T c is presented. The considerations are based on the exact (in classical limit) equations of motion derived by Langer. It is shown that at the initial stages of relaxation the sharp peak appears at the low energy tail of the distribution function. The following time evolution of this peak leads to a delta function formation. When this peak is too narrow the Boltzmann equation is no longer applicable and one must apply the exact equations of motion or the quasi-Boltzmann equation derived in this work which takes into account the coherent effects. The evolution of the distribution function and of the energy spectrum in the system is given by the solution of the equation of motion. The development of a new phase is of explosive character and has a lot of features in common with instabilities in non-linear plasmas, optics, etc., and thus is of quite a general nature. In addition it is shown that the phenomenological models widely used for the description of the dynamics of the lambda -transition in a critical region can be derived from microscopic equations of motion.