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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.
As a result, a nucleus will attract electrons. These electrons will swarm around the nucleus, and the result is an atom. Now we haven't explained everything yet. The electric force explains how the electrons are bound to the nucleus of an atom. But we haven't said anything about what holds the nucleus together. The electric force can't account for this, and in fact, the electric force actually works against holding the nucleus together.