Green functions Schrodinger

A more precise value than in [63] of the nonlogarithmic correction of order a Za) for the IS -state was obtained in [66, 67], with the help of a specially developed perturbation theory for the Dirac-Coulomb Green function which expressed this function in terms of the nonrelativistic Schrodinger-Coulomb Green function [68, 69]. But the real breakthrough was achieved in [70, 71], where a new very effective method of calculation was suggested and very precise values of the nonlogarithmic corrections of order a Zo) for the IS -and 25-states were obtained. We will briefly discuss the approach of papers [70, 71] in the next subsection. 

The solution of single-particle quantum problems, formulated with the help of a matrix Hamiltonian, is possible along the usual line of finding the wave-functions on a lattice, solving the Schrodinger equation (6). The other method, namely matrix Green functions, considered in this section, was found to be more convenient for transport calculations, especially when interactions are included. 

To proceed, we use the relation between the Green function and the eigenfunctions Z a of a system, which are solutions of the Schrodinger equation (6). Let us define Exio) = c in the eigenstate A) in the sense of definition (5), then... 

The average value of any operator O can be written as (O) = (t Os t) in the Schrodinger representation or (O) = (0 Off(t) 0) in the Heisenberg representation, where 0) is some initial state. This initial state is in principle arbitrary, but in many-particle problems it is convenient to take this state as an equilibrium state, consequently without time-dependent perturbation we obtain usual equilibrium Green functions. 

The Green function must satisfy boundary conditions at large distances consistent with the wave function i//. The Schrodinger equation can be replaced by an equivalent Lippmann-Schwinger integral equation... 

The definition of the Schrodinger Green function can be extended to a complexvalued energy parameter z. Then Eq. (7.23) for the Schrodinger equation can be written as... 

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 operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

Compared to the original Schrodinger equation, to which, of course, it reduces for the case = 1, the second term in Eq. (6.8) is new, and corresponds to drift. Again, one can explicitly write down the Green function of the equation with just a single term on the left-hand side. The drift Green function Gd of the equation obtained by suppressing all except this term is... 

This postulate allows the consistent formulation of the Schrodinger, Interaction, and the Heisenberg pictures, the treatment of time dependent perturbations, as well as the description of the quantum events by means of the so-called propagators (Green functions) linking them in a causal ... 

In next, giving its intimacy with wave-function one may whish to establishing the Green function equation, i.e., the analogues of that specific to Schrodinger wave-function equation in coordinate representation ... 

There is clear now that the Green function formalism do not automatically solve the initial Schrodinger problem, but replaces it with a more general one when also the causality of events counts. The passage from the retarded to advanced Green function equation can be easily made though the previously stipulated recipe. 

The continuous electronic states of macroscopic contacts affect the discrete electronic states of isolated molecules calculated by solving the Schrodinger equation thus, a combination of Density Functional Theory and Green function theory (DFT-GF) is used to obtain the electrical properties of the metal-molecule-metal junction. 

The use of Green functions is now widespread in atomic and molecular physics, solid-state theory, nuclear physics, and the theory of many-particle systems in general. At this point we consider the simple problem of the solution of the one-electron Schrodinger equation by Green functions. 

G(x,x I E) is the LDA one-particle Green function which contains the complete sum over excited states and is obtained by solution of the LDA Schrodinger-like equation, ... 

It is, finally, fairly obvious that the method of Green s function and the inhomogeneous Schrodinger equation can also be written down for a system of 3N noninteracting particles, in analogy to the last paragraph in Section 8.9. 

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