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I'm interested in the derivation of the classical Hall effect coefficient, given in cgs by $$R_{H}=-\frac{1}{nec},$$ where $n$ is the electron number density, $-e<0$ is the electron charge,and $c$ is the usual, ubiquitous velocity in Physics, from the fact that QHE provides the quantum of electrical conductance $$g=\frac{2e^{2}}{h},$$ where $h$ is Planck's constant, and the 2 comes from spin degeneracy.

Is there a convenient way to go from the quantum to the classical case for this problem?

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  • $\begingroup$ I can not deliver an answer on the spot, but I can tell you that you should check how many Landau levels remain below the Fermi sea in the magnetic field that you apply. Each of these levels contributes a quantum of conduction to the total current. The higher the magnetic field is, the fewer such conduction channels you have, which corresponds to a higher macroscopic resistance. Hope that helps a bit. $\endgroup$
    – Leviathan
    Jul 15, 2021 at 17:12

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I think you cannot derive classical case from the quantum case. $g=\text{filling factor}\cdot\frac{2e^2}{h}$ occurs at very high magnetic fields where Landau levels start filling and current is carried only by edge states. In classical regime, magnetic fields are low and Landau levels haven't started filling and there is no edge states

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  • $\begingroup$ Any classical experiment can be derived from its quantum mechanical equivalent. Just not the other way around. $\endgroup$
    – Leviathan
    Jul 15, 2021 at 16:59
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If you consider a sample infinite in all directions (no edge states), you will get conventional quantum Hall effect (although it is not classical, since we deal with the Landau levels). This would however require some kind of Kubo-like calculation, so that the current is finite - this is not the case in QHE, where the resistance is finite due to the presence of the confining potential (note that IQHE is but conductance quantization in the quantum point contacts, formed by the sample edges and the magnetic field).

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