This chapter has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. By limiting the orbiting electrons to a series of circular orbits having discrete radii, Bohr could account for the series of discrete wavelengths in the emission spectrum of hydrogen. Light, he proposed, radiated from hydrogen atoms only when an electron made a transition from an outer orbit to one closer to the nucleus. These physical properties (Ve = 174090.938ms-1& Pe = 1.585860888 × 10-25kgms-1) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength (\(\lambda\)0 = 91.12nm) and the Balmer’s experimental observation (\(\lambda\)0 = 91.18nm) for the emission spectrum of hydrogen atom. There are two other constants \(\delta\) = 7.199822 × 10-10 eVm and \(\zeta\) = 1503796277m-1 that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants can be applied to multi-electron systems because they are inherent characteristics of an electron. They can also be derived from the steady state hydrogen atom Schrodinger's equation. For a deeper understanding, these constants may be scrutinised for their determination. The study recommends that current quantum atomic models and equations governing quantum physics should be revisited for better applications and understanding of the physical world.
This chapter has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. By limiting the orbiting electrons to a series of circular orbits having discrete radii, Bohr could account for the series of discrete wavelengths in the emission spectrum of hydrogen. Light, he proposed, radiated from hydrogen atoms only when an electron made a transition from an outer orbit to one closer to the nucleus. These physical properties (Ve = 174090.938ms-1& Pe = 1.585860888 × 10-25kgms-1) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength ((\lambda)0 = 91.12nm) and the Balmer’s experimental observation ((\lambda)0 = 91.18nm) for the emission spectrum of hydrogen atom. There are two other constants (\delta) = 7.199822 × 10-10 eVm and (\zeta) = 1503796277m-1 that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants can be applied to multi-electron systems because they are inherent characteristics of an electron. They can also be derived from the steady state hydrogen atom Schrodinger's equation. For a deeper understanding, these constants may be scrutinised for their determination. The study recommends that current quantum atomic models and equations governing quantum physics should be revisited for better applications and understanding of the physical world.
In this paper, the classical Bohr’s model of the hydrogen atom has been revisited. Two values of fundamental physical properties of an electron in the hydrogen atom has been identified. These physical properties ( & ) are constant in nature. The aim for the review was to contribute to the solution of disagreement between the Bohr’s wavelength ( ) and the Balmer’s experimental observation ( ) for the emission spectrum of hydrogen atom. There are two other constants and that were identified in the Bohr’s equation of the hydrogen atom. The four fundamental physical constants are intrinsic properties of an electron and can be applied to multi-electron system. They can also be obtained from Schrodinger’s equation for hydrogen atom at steady state. These constants may be subjected to scrutiny for their determination for better understanding. Also, since Bohr’s model of hydrogen atom is based on classical mechanics, this paper has provided an alternate method of solving simple problems in atomic physics under Bohr’s model to aid good mental picture of hydrogen atom to scientists.
Abstract The energies, kets and wave functions obtained from the Schrödinger equation for the hydrogen atom are examined in Chapter 9. Three quantum numbers are identified. The energies turn out to be the same as in the Bohr model, and an energy-level diagram appropriate to the quantum description is constructed. Graphs of the probability distributions are interpreted as the electron being in a “cloud” around the proton, rather than at a fixed position: the atom is fuzzy, not sharp-edged. The wavelengths of the five photons of the Balmer series are shown to be in the visible range. These photons are emitted when electrons transition from higher-excited states to the second lowest one, which means that electronic-type transitions underlie the presence of colors in our visible environment. The non-collapse of the atom, required by classical physics, is shown to arise from the structure of Schrödinger’s equation.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
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.
This paper develops and verifies a physics-first description of the hydrogen-atom transition from the L shell (n = 2) to the K shell (n = 1) by combining the historical Bohr model of atomic structure with classical Maxwell electrodynamics. Beginning with the Coulomb force, centripetal dynamics, and Bohr’s angular-momentum quantization condition, the analysis derives the allowed orbital radii, electron velocities, angular momenta, kinetic energies, electrostatic potential energies, and total bound-state energies of the hydrogen atom. For the n = 2 → n = 1 transition, the calculated energy difference is approximately 10.2 eV, corresponding through the Planck–Einstein relation ΔE = hν to a frequency of approximately 2.47 × 10¹⁵ Hz and a wavelength near 121.6 nm, consistent with the hydrogen Lyman-α transition. The emitted radiation is then examined through Maxwell’s equations and the vacuum constitutive relations. The resulting electromagnetic-wave description yields the transverse electric and magnetic fields together with the associated electromagnetic energy density, Poynting energy flux, momentum density, radiation pressure, and Maxwell stress tensor. In this way, the atomic energy difference obtained from the Bohr model is connected quantitatively to the classical propagation properties of the outgoing electromagnetic radiation. The derivation is mathematically consistent as a historical Bohr–Maxwell reconstruction, but it does not constitute a closed dynamical theory of spontaneous emission. The essential bridge between discrete atomic energy levels and the continuous Maxwell field is supplied by the quantum frequency condition ΔE = hν. Neither classical Maxwell electrodynamics nor Newtonian orbital mechanics independently derives this relation, and classical Maxwell theory does not by itself quantize the emitted radiation energy into individual photons. A complete treatment of spontaneous emission, transition probabilities, radiative lifetimes, and photon-field quantization requires quantum mechanics and ultimately quantum electrodynamics. Accordingly, the framework is presented as a verified correspondence between established limiting descriptions rather than as a replacement for Schrödinger wave mechanics, Dirac theory, or quantum electrodynamics. By clearly separating the atomic bound-state calculation, the quantum energy-frequency bridge, and the classical electromagnetic propagation problem, the analysis provides a rigorous conceptual and mathematical foundation for subsequent Maxwell–Schrödinger, Dirac–Maxwell, quantum-electrodynamic, Maxwell–Scretching, and Scretching Quantum Chain formulations of hydrogenic radiative transitions.
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.