Green functions perturbed wave function

There is immediate that having this Dyson series of Green function the wave function solution of the perturbed systems may be written according with the Huygens quantum principle as ... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The fits of Janev et al. [12] stem from a compilation of the results obtained with different theoretical approaches (i) semi-classical close-coupling methods with a development of the wave function on atomic orbitals (Fritsch and Lin [16]), molecular orbitals (Green et al [17]), or both (Kimura and Lin [18], (ii) pure classical model - i.e. the Classical Trajectory Monte Carlo method (Olson and Schultz [19]) - and (iii) perturbative quantum approach (Belkic et al. [20]). In order to get precise fits, theoretical results accuracy was estimated according to many criteria, most important being the domain of validity of each technique. 

G is the reduced Green function of the Schrodinger equation and B = (Us)-Action of the operator O2 on the wave function can be checked not to produce functions more singular than G2 or c2. Therefore, in contrast to the second iteration of the original perturbation, Eq.(12), that of the operator 02 delivers a result which is finite in three dimensions. 

The coefficients of the series may be calculated on the basis of the perturbation theory for the initial- ( i)) and final-state ( /)) wave functions of atom in field. The simplest way of calculations is the use of the iteration procedure for the integral equation with the Coulomb Green function G i(/)(r,r ) ... 

This means that, just as mentioned above about Huygen s principle, every point where the wave function is non-zero at time t = 0 serves as a source for the wave function at a later time. Eq.(3.2) thus describes the propagation of the wave in free space between two different points and times, thereby giving the Green s function its second name the propagator. But what happens if we introduce a perturbing potential We assume that the potential is a point-scatterer (that is, it has no extension) and that it is constant with respect to time, and denote it by 1, which is a measure of its strength. If we now also assumes that we only have one scatterer, located at iq, the time development of the wave becomes ... 

We may say that the range of the Pauli principle is infinity. If somebody paints some electrons green and others red (we do this in the perturbational method), they are in no man s land, between the classical and quantum worlds. Since the wave function does not have the proper symmetry, the corresponding operator + Hg... 

The division of the Hamiltonian into an unperturbed and perturbed part is of course arbitrary. However in most cases jiPq is chosen to make the scattering particle Green s function simple to calculate. Typical choices would be the target plus a free particle or in the case of positive ions, target plus Coulomb wave. Equation... 

---