Green function Schrodinger 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. 

The Green s function for the Schrodinger equation, Eq. (3.2), is defined by the differential equation (Arfken, 1968)... 

To proceed with the proof, we first state a property of the Bardeen integral If both functions involved, i i and x> satisfy the same Schrodinger equation in a region fi, then the Bardeen integral J on the surface enclosing a closed volume w within fl vanishes. Actually, using Green s theorem, Eq. (3.2) becomes... 

For large r, G(f, Fo) must vanish, which requires that A = 0. For small distances, where kf < < 1, it should be identical to the Coulomb potential, which requires that 5=1. Finally, we find that the Green s function of the Schrodinger equation in vacuum is the Yukawa potential. 

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... 

We now turn to approaches that begin with an arbitrary initial function and can, in principle, be iterated to an exact or accurate solution of the SE. The earliest approach is Green s function Monte Carlo (GFMC) in which the time-independent Schrodinger equation is employed [24] DMC was developed later and follows from the time-dependent SE (TDSE) in imaginary time. 

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... 

The fourth method used for quantum chemical calculations is the quantum Monte Carlo (QMC) method, in which the Schrodinger equation is solved numerically. There are three general variants of QMC variational MC (VMC), diffusion QMC (DQMC), and Green s function QMC (GFQMC), all of which... 

The solution of the Schrodinger equation is reduced to solution of the integral equation, well known in the theory of Green s functions ... 

Suppose that we have no potentials in our system, but just free electrons. Then the Hamiltonian is simply Ho = —V2, and the Schrodinger equation can be solved exactly in terms of the free-particle propagator, or Green s function, Go(r, t r, t ), which satisfies the equation ... 

Since the language of the Exact Muffin-Tin Orbital (EMTO) method is phrased in terms of wave functions and the solution to the Schrodinger equation, and not in terms of Green s functions, I will also start with this formulation, and then later make the transition into Green s functions when appropriate. Some papers with a good account of the methodology are Refs.[58, 64, 65]... 

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