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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 quantum approach to the electromagnetic force is called quantum electrodynamics or QED. The electromagnetic force is a force of infinite range which obeys the inverse square law, and is of the same form as the gravity force. The electromagnetic force holds atoms and molecules together. In fact, the forces of electric attraction and repulsion of electric charges are so dominant over the other three fundamental forces that they can be considered to be negligible as determiners of atomic and molecular structure.