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I am trying to understand the hall effects and have a few problems with them.

So let's consider the classical Hall effect. We know that we consider a sample, where the electrons flow, we apply the perpendicular magnetic field, and the electron's motion is curved so they reach the sample's edge. There appears a "new" electric field.

On the other hand, the quantum Hall effect sometimes showed that electrons move in a closed circular orbit, but not in the edge where they can go through "skipping orbits".

So my question is, why in classical hall effect the electrons teachers the sample's edge and don't just move in a circle? What is more, do they move also along the edge like in the quantum hall effect?

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In the classical Hall effect you have cyclotron motion and the charged particles will have this path until they collide with the boundary, then the motion on the boundary becomes a cycloid and charge will continue to pile up on the boundary and this produces a voltage (the hall voltage). Try plotting the path of motion classically on a finite domain.

Now the quantum Hall effect comes in a few forms, for the simplest version you have a the same setup in classical physics only now the path is no longer well defined (you could define a propagator but I digress). You will find for the Integer quantum Hall effect you get landau levels with integer quantization and degenerate energy levels.

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