Categories of existing numerical solution techniques

Almost all interesting quantum problems involve $d\ge2$ degrees of freedom. Almost invariably in this case the wavefunction $\psi$ is represented by a linear sum of basis functions,

$$\psi{({\mathbf r})}\; = \; \sum_{n=1}^N x_n \phi_n{({\mathbf r})}.$$ (5.4)

This is computationally necessary since it reduces the function space of $\psi$ to a finite number $N$ of degrees of freedom, namely the coefficient vector ${\mathbf x}$. For a complete basis, any $\psi{({\mathbf r})}$ can be exactly represented by (5.4) in the limit $N\rightarrow\infty$. Clearly a computer can only handle finite $N$, so the basis should be well chosen so that sufficient convergence is achieved for the $\psi{({\mathbf r})}$ of interest with a manageable $N$. The choice of basis falls into two general classes: (A) The basis does not depend on $E$, or (B) the basis does depend on $E$. This results in two general strategies, summarized in Table 5.1 and now discussed in detail below. The nomenclature ``Class A'' etc which I introduce here is new to the literature.