[noobermin]

There is a way to use physical intuition in QM too, although it obviously isn't the physical kind (tables and flipping them over). It tends to be mathematical, but it certainly isn't rigorous chugging through equations. The canonical example of this are the test questions you see in undergraduate QM courses that show you a space potential graph then ask you sketch the wavefunction in different regions. You have to have an idea what a wavefunction does in certain regions, oscillate? decay? how many nodes, etc.

[reagency]

I can sketch you a graph of (x-1)(x-2)(x-3)(x-4), but it isn't because of intuition. It is because I know how to measure key characteristics of a polynomial (roots and asymptotes) and can smoothly interpolate (simple interpolation is maybe intuitive)

[noobermin]

Sure, but I'm not sure how that is related. What I'm essentially talking about is you have a DE that is of the form -i h\phi_t -b\phi_{xx} + V(x)\phi = 0 and graphing \phi as you vary V(x). That is not obvious unless you know how solutions of the equation behave for different simple cases of V(x) and continuity. That, I argue, is intuition.

Actually, what you mentioned sounds like intuition to me. You didn't make hand plot the polynomial, so that is relying on intuition over rigour, which I think is what the OP's quote from Feynman referred to.