ABSTRACT
It is generally nice to be able to solve a problem exactly. However, this is often not possible, and some approximation scheme must be applied. One example of this is the triangular potential well of Section 2.4 or the finite potential well of Section 2.3.2. In the former example, complicated special functions were required to solve Schrödinger’s equation. In the latter example, the solution could not be obtained in closed form, and graphical solutions, or available software packages, were used to find the eigenvalues of the energy. These are examples of more common problems. In Chapter 6, however, it was pointed out that one could choose an arbitrary basis set of functions that formed a complete orthonormal set as a defining linear vector space. The only requirement on these functions (other than their properties as a set) was that they made the problem easy to solve. In the two previous examples mentioned, what choice would we have made for the set? One choice would have been to use the basis functions that arise from an infinitely deep potential well, but these are not really defined outside the range of the well itself. Nevertheless, it is usually found that it is difficult to find a proper basis set of functions for which the problem can be easily solved.
