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

$$\vert \Psi_i \rangle = \sum_{j}^{I} c_{ij} \vert \Phi_j \rangle$$ (195)

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 one of the available review articles [8,9].