Lech Stawikowski, a student graduating from his high school, wrote an elegant article about solving the Schroedinger equation numerically. He was at that time a participant of the Polish Children's Fund Workshop at the Institute of Theoretical Physics in 2004. Subsequently, he was admitted as a student in the Centre for Interfaculty Individual Studies in Mathematical and Natural Sciences in University of Warsaw. A copy of his article is provided below (in Polish).
The essence of Lech's article is that in order to solve the Schroedinger eigenvalue problem numerically, using a basis of known functions to find the eigenvalues and build the wave functions that solve the problem, one has to decide how many basis functions are needed in order to achieve a desired precision. In simple cases, the decision is not hard to make: by simply increasing the number of functions one eventually achieves the precision one is satisfied with. However, when the problem is more complicated, for example, when it contains two significantly different energy scales, or when it involves cutoff parameters and requires renormalization, it is necessary to estimate the kind and number of basis functions one will need to find a solution. Otherwise, one may engage in expensive, time and memory consuming computations that are in fact futile because the computer imposed limitations exclude that the desired solution can be found.
The bottom-line rule is that the support of the true solution must be covered by the basis functions in both position and momentum space. This condition can be formulated using the concept of turning points where the entire energy of a classical particle becomes equal to the potential energy. This is known in position space. But one also needs to estimate the turning points in momentum space, where the usual kinetic energy plays the role of a new potential energy. The magnitude of the eigenvalue one is seeking provides the order of magnitude of energy for which one needs to find the turning points in position and momentum spaces (representations). The next step is to fill such estimated range with the basis functions. The outcome is an estimate for the number of basis functions with specific parameters that one needs to solve the problem at hand.
For example, if one desires to solve a harmonic oscillator problem with a spring constant k for the state number n, which means for a state with energy E expected to be much larger than the ground state energy, by amount equal about n times the characteristic energy scale of the system (such as the splitting between its two lowest energy levels), using the basis functions of a harmonic oscillator with a spring constant k', then the required number of basis states is N = sqrt(c) n, where c is equal to the larger of the two numbers k/k' and k'/k. Lech's article describes more examples.