\(\cfrac { \partial \psi }{ \partial x }|_{x_f} =0\)
\( \cfrac { \partial ^{ 2 }\psi }{ \partial x^{ 2 } } |_{x_f}=0\)
on the parabola profile. But at the focus, the effect of \(\psi\), and the rate of change of \(\psi\),
\( \cfrac { \partial \psi }{ \partial t } \neq 0\)
We are looking at the solutions of a partial differential equation not an algebraic equation. Strictly speaking,
\( \cfrac { \partial ^{ 2 }\psi }{ \partial x^{ 2 } } |_{x_f}=0\)
cannot be substituted into
\(\cfrac{\partial \psi}{\partial x}\cfrac{\partial \psi}{\partial t}=\cfrac{c^2p}{\sqrt{2}}.\cfrac{\partial^2\psi}{\partial x^2}.e^{-i\pi/4}\)
before solving the equation. But intuitively,
\(\cfrac{\partial \psi}{\partial x}\cfrac{\partial \psi}{\partial t}=0\)
seems valid along \(x=x_f\) and \(\psi\) varies due to a change in \(t\), time only.
