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International Journal of Mathematical Education in Science and Technology

A quantum model of atoms (the energy levels of atoms)

FranÇois Rafie

Published Mar 1, 2001• 1 citation
Publisher

Abstract

In 1913, Niels Bohr developed an accurate model for the hydrogen atom. The mathematics involved and the simplicity of this model is easily understood. The intention of the paper is not to replace Bohr's model, but to find a similar model for all atoms. The model discussed, which was developed on the same basis as Bohr's model for the hydrogen atom, calculates the radii and the energies of the orbits. The wavelength of an emitted photon when an electron makes a ‘jump’ from a higher state to a lower state can also be calculated from the model. Finally, the paper demonstrates that the model obeys the de Broglie's hypothesis that the moving electron exhibits both wave and particle properties.

DOI
10.1080/00207390010010854

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In 1913, Niels Bohr developed an accurate model for the hydrogen atom. The mathematics involved and the simplicity of this model is easily understood. The intention of the paper is not to replace Bohr's model, but to find a similar model for all atoms. The model discussed, which was developed on the same basis as Bohr's model for the hydrogen atom, calculates the radii and the energies of the orbits. The wavelength of an emitted photon when an electron makes a ‘jump’ from a higher state to a lower state can also be calculated from the model. Finally, the paper demonstrates that the model obeys the de Broglie's hypothesis that the moving electron exhibits both wave and particle properties.
Bohr ignored the electromagnetic interaction of moving charges in atoms and only considered Coulomb forces, thus encountering half frequency difficulties. This article considers the Lorentz force to make the classical Bohr model consistent with modern quantum theory, uses de Broglie's standing wave principle instead of Bohr's quantization assumption, derives the energy level formula for hydrogen atoms, and explains the radiation mechanism of hydrogen atomic spectra. The hydrogen atom in a spectral tube emits a photon wave train during one free path, and the wavelength of the photon is the eigen wavelength of its wave train. The head wavelength of the wave train is four thirds of its eigen wavelength, and the tail wavelength of the wave train is two-thirds of its eigen wavelength.
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.
The atom model of Niels Bohr is commonly considered as antiquated even if it describes the atomic spectrum of hydrogen quite accurately. The later published relation of Louis De Broglie could arithmetically be implemented into Bohr’s formulation, leading to the concept of standing waves as the existence cause of excited electron states. However, at that time it was not possible to find well defined electron trajectories being classically describable. As a consequence, the »quantum mechanics« were developed by several authors, particularly by Schrodinger and by Heisenberg, but delivering a hardly under¬standable formalism where the classical physical laws appear being abrogated while abstract terms replace concrete and imaginable ones, abandoning the particle perception of mass points. In particular, Heisenberg’s »uncertainty principle« and the assumption of state probabilities seem to be in a striking variance to the idea of standing waves. In contrast, a formulation is given here which exactly describes the electron trajectories in the exited states solely by applying classical physical laws. Firstly, the original Bohr-model - in combination with De Broglie’s relation - is rolled up. From this formula system a vibration frequency - corresponding to the De-Broglie frequency – is deduced which is n-times larger than the rotation frequency of the Bohr-model. Furthermore, a direct coherence between that vibration-frequency of the electron and the frequency of the involved light is evident being explainable as a resonance effect. Then, a three-dimensional model is proposed where the electron oscillates and pulses perpendicularly to a virtual rotation plane i.e. rotating around a vertical axis, accompanied by a perpetual energy exchange between potential and kinetic energy. This leads at the excited, metastable energy states to well-defined, three-dimensional and wavy trajectories winding up on a surface similar to the one of a hyperboloid, whereas at the ground state the trajectory is planar and stable.
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 (λ\lambdaλ0 = 91.12nm) and the Balmer’s experimental observation (λ\lambdaλ0 = 91.18nm) for the emission spectrum of hydrogen atom. There are two other constants δ\deltaδ = 7.199822 × 10-10 eVm and ζ\zetaζ = 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.
Turning back to Niels Bohr's model of the hydrogen atom, the paper is trying to resolve one of the issues associated with the model.While N. Bohr in the framework of his model did not find any relation between electron's motion in the atom and radiation emitted, there is a reason to believe that the relation is bound to exist.Furthermore, it is already given by Rydberg's formula that expresses the frequency of radiation as a difference of two other frequencies associated in some manner with electron's rotation.Based on this approach a new value for the radius of the ground stationary orbit is calculated.It appears to be more than Bohr's radius by a factor of 1.588.The question why both values for the radius of the ground stationary orbit (Bohr's radius and a new one) are compatible with Rydberg's formula is considered.Finally, what exactly Niels Bohr overlooked in his model of the hydrogen atom, is discussed.It is established that Rydberg's formula can be derived from the more general parameterized postulates, when Niels Bohr used, regardless of the value of the parameter.
The existence of line spectra can be explained bymeans of the first “quantum”model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra. The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.