Matrix Mechanics

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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 $\hat{H} \psi_e({\bf x}) = E_e \psi_e({\bf x})$ as a matrix problem, we need to find a suitable linear vector space. Now $\psi_e({\bf x})$ is an N-electron function that must be antisymmetric with respect to interchange of electronic coordinates. As we just saw in the previous section, any such N-electron function can be expressed exactly as a linear combination of Slater determinants, within the space spanned by the set of orbitals $\{ \chi({\bf x}) \}$ . If we denote our Slater determinant basis functions as $\vert \Phi_i \rangle$ , then we can express the eigenvectors as

$\begin{displaymath}\vert \Psi_i \rangle = \sum_{j}^{I} c_{ij} \vert \Phi_j \rangle \end{displaymath}$

(167)

for I possible N-electron basis functions (I will be infinite if we actually have a complete set of one electron functions $\chi$ ). Similarly, we construct the matrix ${\bf H}$ in this basis by $H_{ij} = \langle \Phi_i \vert H \vert \Phi_j \rangle$ .

If we solve this matrix equation, ${\bf H} \vert \Psi_n \rangle = E_n \vert \Psi_n \rangle$ , 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 $\chi$ . If we restrict the N-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 the review article by Shavitt [8].

Next: Bibliography Up: Solving the Electronic Eigenvalue Previous: The Nature of Many-Electron