Hmmm, why should F = ma? You've made me wonder that now!! Out the window goes quantum! Do you never wonder why any of the stuff they teach us!!?
An accelerating charge creates an electromagnetic wave -- it's a consequence of Maxwell's equations, I think Duffin is a book which shows why.
F=ma, cos we choose what we want acceleration to be, we choose how we define F, so in fact it is all the more surprising that we choose some fairly random definitions of these quantities, and they obey some other laws which we guess. We choose what a wavefunction should be, we choose what energy means, we choose what position, momentum is, so it is surprising that it all works.
But we're not given the full beauty of it. For example, F=ma and all that is a rubbish way to analyse systems. Much better is to define the Lagrangian of a system as L=T-V, where T is the kinetic energy, V is the potential energy.
Then you say that the time integral of L along some path is S, the action, namely: S=∫x1x2Ldt. Then you use the principle of stationary action (which is often misquoted as the principle of least action or principle of least effort) which says that the system will move from x1 to x2 so that the functional derivative of S with respect to the path is 0.
And then if you slightly redefine S, and the integral instead of taking one path from x1 to x2, to taking all paths from x1 to x2 (and also change the time-space dependence so that all variables are treated equally) you get Feynman's path integral formulation of quantum mechanics, which also works just as well as the Schrodinger formalism.