Pre-Quantum Electrodynamics

We can provide a very precise statement about uniqueness of solutions to Poisson's (or Laplace's) equation with some very basic considerations starting from the divergence theorem \[ \int_{\cal V} d\tau ~{\boldsymbol \nabla} \cdot {\bf F} = \oint_{\cal S} da ~{\bf F} \cdot {\bf n} \] Let \({\bf F} = \phi {\boldsymbol \nabla} \psi\), where \(\phi\) and \(\psi\) are scalar fields. We can then write \[ {\boldsymbol \nabla} \cdot (\phi {\boldsymbol \nabla} \psi) = \phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi \] and \[ \phi {\boldsymbol \nabla} \psi \cdot {\bf n} = \phi \frac{\partial \psi}{\partial n}. \] Substituting this in the divergence theorem gives Green's first identity

\[ \int_{\cal V} d\tau ~(\phi {\boldsymbol \nabla}^2 \psi + {\boldsymbol \nabla} \phi \cdot {\boldsymbol \nabla} \psi) = \oint_{\cal S} da ~\phi \frac{\partial \psi}{\partial n}. \tag{Green1}\label{Green1} \] This first identity will prove crucial in the argument that follows. As an aside for now, for completeness, if we do the same thing again but with \(\phi\) and \(\psi\) interchanged, and subtract the result, we obtain another useful result known as Green's second identity or Green's theorem

\[ \int_{\cal V} d\tau (\phi {\boldsymbol \nabla}^2 \psi - \psi {\boldsymbol \nabla}^2 \phi) = \oint_{\cal S} da \left(\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right). \tag{Green2}\label{Green2} \]