This chapter describes quantum mechanically the electrons in the Fermi Sea and obey the Pauli Exclusion Principle. It also describes free one-dimensional electron gas quantum mechanically. Electrons are subject to external forces due to electric and magnetic fields. Electrons also respond to magnetic fields and it seems natural to investigate their motion in the presence of both of these fields. The quantum Hall effect pertains to Klaus von Klitzing’s discovery that for very clean samples and large magnetic fields, the transverse or Hall resistivity exhibits plateaus at specific field values. The classical transverse resistivity is the expected straight line behavior with the magnetic field and, finally, some constant is chosen for the classical longitudinal resistivity. The description of the prize reads “The three researchers are being awarded the Nobel Prize for discovering that electrons acting together in strong magnetic fields can form new types of ‘particles’, with charges that are fractions of electron charges.”.