Quantized Hall resistance and the measurement of the fine-structure constant

Abstract

An elementary, exact calculation of two-dimensional electrons in crossed electric and magnetic fields with a $\ensuremath{\delta}$-function impurity is carried out in the quantum limit. A state localized on the impurity exists and carries no current. However, the remaining mobile electrons passing near the impurity carry an extra dissipationless Hall current exactly compensating the loss of current by the localized electron. The Hall resistance should thus be precisely $\frac{h}{{e}^{2}}$, as found experimentally by Klitzing et al. Other possible sources of deviation from this result are briefly examined.