Semiconductors
Every chip in every phone, laptop, and computer relies on band theory — the quantum-mechanical description of how electrons occupy discrete energy bands in a crystal, with gaps between them that ordinary electrons can't cross. Silicon's particular gap size is exactly what makes it useful as a semiconductor: small enough to control with a little added energy, large enough to switch cleanly on and off.
Transistors, the on/off switches that make up every digital circuit, are a direct engineering application of this quantum band structure.