Distribution of electron orbits having a definite angular momentum in a static magnetic field

Abstract

Electron orbits having a definite angular momentum in a static magnetic field are calculated with the aid of the Bohr-Sommerfeld quantization rules. The quantization gives that orbits are arranged along a straight line but the distance between the centers of two neighboring orbits decreases with increase of the absolute value of the angular momentum. With the energy correction equal to the zero-point energy of the harmonic oscillator, the distribution of orbits becomes identical to that obtained recently with the aid of a mixed semiclassical and quantum mechafical theory. A well-known semiclassical approach started by Onsager [ 31 allowed us to quantize the flux of the magnetic field together with the electron orbital motion. More recently, the quantum Hall effect raised a problem of the filling factor of the Landau levels, the origin of which is the degeneracy of these levels. The filling factor of the quantum levels in a constant magnetic field plays an essential role in both the integer and the fractional quantum Hall effects [41.Electron orbits are a very convenient picture of the electron behavior, but they have been traced precisely by the old quantum theory in only few cases 15, 61. Perhaps the best-known example are electron orbits in the hydrogen atom where circular orbits were obtained for the quantum number n = k at the distances n2h2 m,e2 y = -from the nucleus; here, n is a principal quantum number, and k, an azimutal quantum number. For