My only exposure to energy bands is coming from courses on semiconductor devices (diodes and transistors), and I only understand the very basic theories of quantum mechanics (I've only taken a course which briefly introduced it). The thing is, we keep using energy band diagrams to explain and derive the properties of diodes and transistors in my circuits courses, so I'm really trying to understand them. Band bending is one thing that gets used a lot, and I can't seem to make sense of it. From what I understand, each discrete energy level in the band corresponds to some particular energy state that only two electrons may occupy because of the Pauli exclusion principle. This is why I don't understand why it makes sense to "bend" the energy band. Say there's only one electron occupying an energy level. When the energy level bends (e.g. due to an applied electric field), does that mean the electron's energy is now position dependent? Can it "slide" along that energy level, gaining/losing energy in a continuous fashion? That doesn't seem right to me based on what I've seen of quantum mechanics where particle energy always seem to be quantized, but again, my knowledge is very limited in that sense. I guess I just don't really understand why the energy of a given energy level would be position dependent.