Is electron energy spectrum quantized in the presence of an electric field? Maybe this is a stupid question, but is free electron's energy spectrum continuous in the presence of an external field? For example, if a free electron flies between two plates in a capacitor.
 A: Boundary conditions create a discrete spectrum. A potential well creates boundary conditions. you can see this illustrated at Hyperphysics.
If an electron is confined to a box with an infinite potential outside, the solution is sinusoidal inside the well and $0$ outside. The wave function is continuous, so the value must be $0$ at the edge. This implies a there must be a half integer number of wavelengths inside, which implies a discrete energy spectrum. 
This is the most obvious example. But the same thing applies to other potential wells. For a box with a finite potential, the solution is sinusoidal where the potential is lower than the electron's kinetic energy, and a decaying exponential where it is higher. The sinusoids must meet the decaying exponentials smoothly. 
A parallel plate capacitor is like a potential well in the z direction, and like a free particle in the x and y directions. The potential linearly increases away from the + plate, and is constant in x and y. For an electron with low kinetic energy, there is a height where the potential is bigger. The solution is sinusoidal below and a decaying exponential above. 
The boundary condition at the bottom plate is a little trickier. You would have to have an insulating layer so the electron wouldn't simply be absorbed by the bottom plate. Given that, you could model it like a potential barrier. 
You could find a time independent solution where the electron stays near the bottom plate, but travels freely horizontally.
A 1 dimensional problem like this would have a discrete spectrum. But a particle that travels freely in x and y can have a continuous spectrum. 
