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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.
And the reason they don't is because of the strong nuclear force. In physics, there are four fundamental forces: gravity, the electromagnetic force, the weak force, and the strong force. The strong force is, unsurprisingly, strong. The strong force is the force that holds the nucleus of the atom together, even through other forces want to pull it apart. The electrostatic repulsion (part of the electromagnetic force) between the positive protons in the nucleus is very powerful.