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## Matrix Mechanics

As we mentioned previously in section 2, Heisenberg's matrix mechanics, although little-discussed in elementary textbooks on quantum mechanics, is nevertheless formally equivalent to Schrödinger's wave equations. Let us now consider how we might solve the time-independent Schrödinger equation in matrix form.

If we want to solve as a matrix problem, we need to find a suitable linear vector space. Now is an -electron function that must be antisymmetric with respect to interchange of electronic coordinates. As we just saw in the previous section, any such -electron function can be expressed exactly as a linear combination of Slater determinants, within the space spanned by the set of orbitals . If we denote our Slater determinant basis functions as , then we can express the eigenvectors as

 (195)

for possible N-electron basis functions ( will be infinite if we actually have a complete set of one electron functions ). Similarly, we construct the matrix in this basis by .

If we solve this matrix equation, , in the space of all possible Slater determinants as just described, then the procedure is called full configuration-interaction, or full CI. A full CI constitues the exact solution to the time-independent Schrödinger equation within the given space of the spin orbitals . If we restrict the -electron basis set in some way, then we will solve Schrödinger's equation approximately. The method is then called configuration interaction,'' where we have dropped the prefix full.'' For more information on configuration interaction, see the lecture notes by the present author [7] or one of the available review articles [8,9].

Next: Bibliography Up: Solving the Electronic Eigenvalue Previous: The Nature of Many-Electron   Contents
David Sherrill 2006-08-15