How did the understanding of quantum physics lead to the development of the microwave oven?

The development of the microwave oven isn't really an extension of quantum mechanics. It's more an extension of the work of Faraday, Maxwell and Hertz, and that work was done in the late 1800's. The microwave oven applies electromagnetic radiation (emr) to the heating of stuff.

And emr has everything to do with electricity and moving electrons. It really doesn't have that much to do with anything relating to the nucleus of the atom. That's generally where quantum mechanics is at its best.

Though quantum mechanics gave us a much better "look" at the electron, its basic nature as regards what we can do with it to generate emr doesn't really require knowledge of, say, an electron's quantum mechanical property of spin. We build a magnetron and generate microwaves to pop popcorn and (re)heat coffee without any help from giants like Werner Heisenberg, Max Planck, Louis de Broglie, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli, Richard Feynman and others who spent the largest portion of their lives - their lives contemplating and defining the quantum mechanical nature of the world in which we live.

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