Heisenberg equations of motion for, the Coulomb problem and so(4.2)

Owen L. de Lange, R. E. Raab

Shin-ichiro Kondo

To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green's function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.

D. Dubbers, H.‐J. Stöckmann

Z. Ficek, Mohamed Ridza Wahiddin

In the master equation method, we have already illustrated a powerful technique for the calculation of the dynamics of an atomic system interacting with the vacuum field. Another technique for calculating the dynamics of an atomic system coupled to the elec- tromagnetic (EM) field involves Heisenberg equations of motion for the system’s operators. A difference between the master equation and the Heisenberg equations is that the later involves dynamics of the operators, which allows to analyse the evolution of an atomic system in terms of the field and atomic operators. This creates some problems with handling the Heisenberg equations as, in general, operators do not commute and then in the course of solution of the equations we may face the problem of ordering of the operators. It is usually resolved by putting the operators in the normal order. We have gained some experience with the Heisenberg equations of motion in Chapter 6, Example 6.3, where we studied squeezing generation in the nonlinear degenerate parametric amplifier (DPA) process. Here, we illustrate the technique on the standard model of a two-level atom interacting with a multi-mode field. We then generalize the technique to some specific models such as Lorenz- Maxwell and Langevin equations, the derivation of which involves some approximations that can be applied only in some limited cases. We also present in detail the Floquet approach, which is usually applied to problems determined by differential equations with time- dependent coefficients.

Richard L. Liboff