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Propagators in Quantum Chemistry, 2nd Edition by Yngve Öhrn, Jan Linderberg

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Chapter 2

Differential Equations and Green’s Functions

Consider a homogeneous differential equation in one dimension on axb

Image

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.,

Image

and for a = 0, the solutions Image are obtained

Example 2: Harmonic oscillator, i.e.,

Image

with the solutions Image where Image 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 ...

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