Propagators in Quantum Chemistry, 2nd Edition

Chapter 2

Differential Equations and Green’s Functions

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 Φ_n (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 H_n 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 ...