|MadSci Network: Physics|
Very Good Question!
You'll hardly ever find anyone discuss transistors in quantum terms, because the quantum effects give rise to classical-looking physics which is much easier to talk about. But all of the classical-looking physics is really based on quantum mechanics, and without an understanding of QM it is extremely unlikely that anyone would have invented the Field Effect Transistor.
Classically, there is no reason to think that an electron could move through a semiconductor, with all those close-packed atoms in the way. Even if it could leak through, you might (classically) expect electrons to dribble out exponentially, losing energy in the process, rather than moving along like particles with well-defined energies. The effective mass of electrons in semiconductors is not the electron mass, but rather 2-20 times this value; the actual value depends on Planck's constant and quantum amplitudes. Classically there's no such thing as a "hole", and classically it wouldn't be obvious how doping semidonductors would have any effect on the electrons. The very idea of Fermi levels and conduction bands come straight from quantum.
But the amazing thing is, when you go through all the quantum calculations, you get classical-looking results! Once you understand the underlying quantum mechanics, you can pretend you've found "new" classical laws of physics which only hold in semiconductors, in which there are different- mass electrons, real particles called holes, and doping effects. Usually transistor explanations are couched in this "classical" way of thinking, even though all of the supposedly classical laws are all based on quantum mechanics.
The best discussion I've seen of semiconductors from first-principles (quantum mechanics) is in the Feynman Lectures on Physics (Vol III, chpts 13-14). Very interesting reading.
Nowadays, people are starting to make transistors with additional quantum effects (because the transistors are so small that new effects come into play.) One example is the quantum dot transistor. But that doesn't make all other transistors purely classical.
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