Consider a homogeneous differential equation in one dimension on a ≤ x ≤ b
In the study of stationary states in quantum chemistry, one would normally introduce boundary conditions, as for instance, Φ (a) = Φ (b) = 0 and solve the resulting eigenvalue problem. Solutions occur only for certain values of E = ∊n, so-called eigenvalues, and the corresponding solutions Φn (x) are called eigenfunctions.
Example 1: Particle in a box, i.e.,
and for a = 0, the solutions are obtained
Example 2: Harmonic oscillator, i.e.,
with the solutions where and Hn is a Hermite polynomial of degree n.
The previous two example solutions were obtained by traditional solution methods, as, for example, a series method. Instead of proceeding in this manner, we consider a general solution of the second-order differential equation (2.1):
where u and v are particular ...