Schrodinger inhomogeneous

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. 

O for its orthogonal complement. Instead of the original Schrodinger equation (1.1), we will now study the inhomogeneous equation... 

In the previous section 4 the exact formulation of perturbation theory has been considered, i.e. it has been assumed that both the unperturbed Schrodinger equation and the inhomogeneous equations of the various orders of perturbation theory are solved exactly. This is hardly realized in... 

From Eq. (67), it is easy to see that A+) satisfies the inhomogeneous Schrodinger equation... 

The superscript S indicates the nonrelativistic Hartree-Fock approach based on the Schrodinger one-electron operator. The Hartree-Fock equations are written in a somewhat artificially homogeneous fashion because of the devision of X (r) by Pj(r), and inhomogeneities arise from the y-shell contributions through Pj(r) (and therefore the Hartree-Fock interaction potentials are pseudo-local). Potential singularities due to the nodes of the Pj(r) functions are ignored. 

An inhomogeneous equation, whose solution is analogous to that of the time-independent Schrodinger equation. The inhomogeneity reflects the preparation of the system at time t = 0 in the wave packet state x(t = 0)). 

Rewriting the inhomogeneous version of the Schrodinger equation, where the boldface wave vectors below signify added dimensions, one obtains (the poles of the first line of Eq. 1.8 correspond to eigenvalues below)... 

The treatment of Schrodinger s perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. 

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