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# Differential Equations and Green’s Functions

Consider a homogeneous differential equation in one dimension on axb 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 ...

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