Approximation by the Use of Difference Equations

—The approximation to a wave function if by segments of straight lines. 

Just as the lowest energy W of the differential equation can be obtained by minimizing the energy integral E = f j) H t dT with respect to the function 0, keeping /0 / dr = 1, so the lowest value of W giving a solution of Equations 27-40 may be obtained by minimizing the quadratic form 

A convenient method1 has been devised for carrying out this minimization. A set of trial values of t is chosen and the value of E is calculated from them. The true solutions 4 i, to which the values of fa will converge as we carry out the variation, satisfy Equations 27-40. Transposing one of these gives 

If the fa s we choose are near enough to the true values ipi, then it can be shown1 that, by putting , i and ,+i in place of and pi+i and E in place of W in Equation 27-44, the resulting expression gives an improved value for f i, namely, 

This procedure may be modified by the use of unequal intervals, and it can be applied to problems in two or more dimensions, but the difficulty becomes much greater in the case of two dimensions. 

---