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So the electromagnetic force holds matter together, but in a quantum mechanical way. As for metallic bonds I have no clue how that works. The idea seems to be that a sea of electrons acts as a glue that holds together an array of nucleons. Not sure how Schrodinger's equation gets you that.
The Strong Force. A force which can hold a nucleus together against the enormous forces of repulsion of the protons is strong indeed. However, it is not an inverse square force like the electromagnetic force and it has a very short range. Yukawa modeled the strong force as an exchange force in which the exchange particles are pions and other heavier particles. The range of a particle exchange force is limited by the uncertainty principle. It is the strongest of the four fundamental forces.