Bohr atomic model is based on the assumption that electrons on allowed quantized orbits do not radiate. A quantum mechanical justification of this hypothesis, based on the Madelung Bohm treatment of the Schr\odinger equation is presented.
The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.
One of the most important developments of modern science was the emergence of the concept of energy quantization which forced its way into our previous classical understanding of matter in nature. Modern day view of matter constituents and present understanding of atomic structure basic to quantum chemistry arose in those first decades of last century. In this line, a key feature for the understanding of microscopic atomic structures and their stability was the concept of angular momentum quantization. To capture the beauty and the power of these concepts is therefore extremely relevant for apprehending the contemporary view of matter, energy and interactions in nature. In this work we present a possible alternative approach of quantizing the atomic orbit for the hydrogen's atom model proposed by Bohr without using his hypothesis of angular momentum quantization from the start. In our approach here we show that when we identify the number of photons and the quantum number for the orbit, we are led to Bohr's hypothesis as a consequence of the Planck's energy quantization.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.