Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

Comparison of Bohr and Hulthén hydrogen atomic energy levels using the time-independent solution of Schrodinger’s equation and confluent hypergeometric function

Guruh Sukarno Putra, Hamdi Akhsan

Published Jan 1, 2022• 0 citations
Publisher

Abstract

The energy for each level of the hydrogen atom was triggered by Neils Bohr, where the amount of energy decreases at every level of orbit. The value of energy depends on the principal quantum number which only applies to atoms with the s subshell of an atom with the same characteristics as the hydrogen atom. With the advancement of science, specifically in quantum physics, many potential equations aimed to solve the time-independent Schrodinger equation with the solution of the estimated value of energy. One of the applicable potential equations is the Hulthen potential which is explained about the potential at a small distance and only applies to atoms and micro-scale objects in physics and only has a solution at the azimuth quantum number value l = 0 or groups of atoms subshell s. Using the confluent hypergeometric function, the combination of the Hulthén potential and time-independent Schrödinger equation generates the estimated energy value of the hydrogen atom level, which can be used as a comparison with the estimated energy value of Bohr’s hydrogen atom. This study found that the comparison between Hulthén and Bohr energy level lies in the same ratio 1:1

DOI
10.1063/5.0072413

Related papers

Why is the Solution of the Schrödinger Equation not a Miracle in Quantum Chemistry? A Perspective From the Heisenberg Uncertainty Principle

Lieu Le Quang, Liviu Giurgiulescu, Khiem Tran Ngoc +1 more

The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.

Technical Education Science/Giáo dục Kỹ thuật•Sep 17, 2026•0 citations

Bohr’s Model of Hydrogen Atom: A Review

Robert Wilson

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.

Book Publisher International (a part of SCIENCEDOMAIN International)•Jun 8, 2023•0 citations

Solving the Schrödinger equation for Kosen students : Hydrogen atom

祐二 佐々井

Solving the Schrödinger equation in quantum mechanics is not easy. In this research report, we would like to solve the Schrödinger equation concretely for the energy eigenvalue problem of the hydrogen atom. To help Kosen students understand this problem, we explain general solutions from the calculation of concrete solutions. We also explain the relationship between the principal quantum number n, the azimuthal quantum number l, the magnetic quantum number m, and the spin quantum number s.

•Mar 30, 2021•0 citations

Chapter 27: Bohr’s Model of Hydrogen Atom

Muzakir Saifful Kamaluddin

Neils Bohr suggested several important postulates in order to describe the behavior of electron in hydrogen atom. Bohr formulated an equation to estimate the minimum radius of a hydrogen atom can have and the energy level of electron \nbased on their orbit.

•Jan 1, 2011•0 citations

Hydrogen Atom

Ram Yatan Prasad, Pranita

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).

CRC Press eBooks•Jan 15, 2021•0 citations

Wrong Potential Energy Term in Schrödinger’s Equation for Hydrogen Atom

Gurcharn Singh Sandhu

Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.

Journal of Modern Physics•Jan 1, 2018•3 citations
The energy for each level of the hydrogen atom was triggered by Neils Bohr, where the amount of energy decreases at every level of orbit. The value of energy depends on the principal quantum number which only applies to atoms with the s subshell of an atom with the same characteristics as the hydrogen atom. With the advancement of science, specifically in quantum physics, many potential equations aimed to solve the time-independent Schrodinger equation with the solution of the estimated value of energy. One of the applicable potential equations is the Hulthen potential which is explained about the potential at a small distance and only applies to atoms and micro-scale objects in physics and only has a solution at the azimuth quantum number value l = 0 or groups of atoms subshell s. Using the confluent hypergeometric function, the combination of the Hulthén potential and time-independent Schrödinger equation generates the estimated energy value of the hydrogen atom level, which can be used as a comparison with the estimated energy value of Bohr’s hydrogen atom. This study found that the comparison between Hulthén and Bohr energy level lies in the same ratio 1:1
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
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.
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).
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.