Green function quantum systems

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. 

Quantum field theory provides an unambiguous way to find energy levels of any composite system. They are determined by the positions of the poles of the respective Green functions. This idea was first realized in the form of the Bethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)... 

The technical problem is of course to develop an adequate form of the intersubsystem junction for the case when the quantum system is represented by the d-shell. This is done using the EHCF(L) technique described above. In the EHCF(L), the effective crystal field in agreement with the general theory of Section 1.7.2 is given in terms of the /-system Green s function. The natural way to go further with this technique is to apply the perturbation theory to obtain estimates of the /-system Green s function entering eqs. (4.83) and/or (4.92). That is what we shall do now. 

This statement implies that not only the Coulomb interaction is included in Er and Exc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessruily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the noiu-elativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take... 

For both bosonic systems and fermionic systems in the fixed-node approximation, G has only nonnegative elements. This is essential for the Monte Carlo methods discussed here. A problem specific to quantum mechanical systems is that G is known only asymptotically for short times, so that the finite-time Green function has to be constructed by the application of the generalized Trotter formula [6,7], G(r) = limm 00 G(z/m)m, where the position variables of G have been suppressed. 

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 ... 

Buendia et al. calculated many states of the iron atom with VMC using orbitals obtained from the parametrized optimal effective potential method with all electrons included. Iron is a particularly difficult system, and the VMC results are only moderately accurate. The same authors also pubhshed VMC and Green s functions quantum Monte Carlo (GFMQ calculations on the first transition-row atoms with all electrons. GFMC is a variant of DMC where intermediate steps are used to remove the time step error. Cafiarel et al. presented a very careful study on the role of electron correlation and relativistic effects in the copper atom using all-electron DMC. Relativistic effects were calculated with the Dirac-Fock model. Several states of the atom were evaluated and an accuracy of about 0.15 eV was achieved with a single determinant. ... 

In the present article we review the theory of Green s functions and propagators. We discuss which properties of a quantum system and which physical processes can be described by these functions. A general approximation scheme to evaluate Green s functions and propagators is briefly outlined and the computational aspects involved in their calculations are discussed. Particular attention is paid to the one-particle Green s function and to the particle-particle propagator as representative examples. Several illustrative applications are presented and related to experiment. 

J. Cizek and J. Paldus, A direct calculation of the excitation energies of closed-shell systems using the Green function technique, Int. J. Quantum Chem. Symp. 6 435 (1972) discussion of the status of large-molecule quantum chemistry, in Energy, Structure, and Reactivity , D. W. Smith and W. B. McRae, eds., Wiley, New York (1973), p. 389. 

The fermion and Green s Function Monte Carlo are important in themselves and interesting as hints to the richness of methodology that can be brought to bear on the computation of quantum systems. In the short term we expect to broaden the specific trial functions used in fermion Monte Carlo to permit the more accurate study of He-3 and the treatment of more realistic models of nuclear and neutron matter. We expect also to try a variant in Quantum Chemistry problems. 

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