Green’s Functions as Thought Experiments

Abstract

Green’s theorem

$$ \int_V {(\Phi \nabla ^2 \Psi - \Psi \nabla ^2 \Phi )d^3 x = } {\text{ }}\int_S {(\Phi \overrightarrow \nabla \Psi - \Psi \overrightarrow \nabla \Phi ) \cdot \overrightarrow n dS} $$

((4.1))

(where $\vec n$ is the outgoing normal to the surface S bounding the volume V) is obtained from the divergence theorem

$$ \int_V {div\overrightarrow A d^3 x = } {\text{ }}\int_S {\overrightarrow A \cdot \overrightarrow n dS} , $$

((4.2))

with $\overrightarrow A = \Phi \overrightarrow \nabla \Psi - \Psi \overrightarrow \nabla \Phi$. Consider the one-particle Schrödinger equation

$$ ( - \frac{1} {2}\nabla _{\overrightarrow r }^2 + V(\overrightarrow r ) - \varepsilon {\text{)}}\psi {\text{(}}\overrightarrow r {\text{) = 0}} $$

((4.3))

defined in some volume V with some boundary conditions; it is often convenient to change it into an integral equation by a method which is familiar from classical physics. One introduces a Green’s function satisfying

$$ ( - \frac{1} {2}\nabla _{\overrightarrow r }^2 + V(\overrightarrow r ) - \varepsilon {\text{)}}G{\text{(}}\overrightarrow r ^\prime {\text{,}}\overrightarrow r {\text{) = }}\delta {\text{(}}\overrightarrow r - \overrightarrow r ^\prime {\text{),}} $$

((4.4))

multiplies (4.3) by $(\overrightarrow r ^\prime ,\overrightarrow r ^\prime )$, (4.4) by $\psi (\overrightarrow r ^\prime )$ and subtracts; the result is (exchanging $\overrightarrow r$ with $\overrightarrow r ^\prime$)

$$ \psi (\overrightarrow r ) = \frac{1} {2}\int_V {d^3 \overrightarrow r ^\prime \left[ {G{\text{(}}\overrightarrow r - \mathop {r'}\limits^ \to {\text{)}}\nabla _{\overrightarrow r ^\prime }^{\text{2}} \psi (\overrightarrow r ^\prime ) - \psi (\overrightarrow r ^\prime )\nabla _{\overrightarrow r ^\prime }^{\text{2}} G{\text{(}}\overrightarrow r - \mathop {r'}\limits^ \to {\text{)}}} \right]} \cdot $$

((4.5))

The V integral can be changed to a surface integral by Equation (4.1).