I remember watching a documentary about early works in quantum physics and how it was essential to the microelectronics revolution and the invention of the transistor. What part exactly helped? Is there a source on technical aspects that lead to such development?
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One word: Semiconductors.
Semiconductor physics cannot be explained using classical physics. One needs to understand the electronic band structure of these materials in order to engineer various devices (e.g., p-n junctions, p-n-p junctions etc.) using them. Without such basic devices we won't be able to engineer modern electronic devices.