The Convergence of Semi-Classical and Quantum Mechanics: Examining Atomic Radiation through the Lens of the Correspondence Principle

Abstract

The semi-classical, or Bohr-Sommerfeld, model of a hydrogen atom concerns a "de Broglie" electron in an elliptical orbit in the Coulomb-law field of a proton. This model led to the later quantum mechanical analysis of Schrodinger and Heisenberg, whose equations model electrons with a probabilistic wave function. Semi-classical and quantum approaches have their own derivations of parameters describing the properties of the atom from quantized energy structure to radiated power. Despite the differences in physical origin between the models, the correspondence principle formulated by Bohr states that in the limit of large quantum numbers, semi-classical and quantum descriptions merge. This thesis considers the comparison of quantum and semi-classical models for frequency and the power radiation of the electronically excited hydrogen atom. Using the mathematics as a base, this paper analyzes correspondence between the semi-classical and non-relativistic quantum models as quantum numbers are increased. It is shown that in the limit of large quantum numbers, there is correspondence for both the frequency and the radiated power of the excited hydrogen atom. Knowing classical mechanics and the de Broglie hypothesis is sufficient indeed to anticipate the qualitative and, in some cases, quantitative decay of highly excited atomic hydrogen.