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

CRC Press eBooks

The Hydrogen Atom

Edward K. Yeargers

Published Jan 18, 2018• 0 citations
Publisher

Abstract

The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.

DOI
10.1201/9781351070072-12

Related papers

From the Schrödinger Cloud to the Bohr Orbit: A Sequential Confinement Approach to the Hydrogen Atom

Ricardo Ugarte

The Bohr model of the hydrogen atom, despite its heuristic success in predicting discrete energy levels, is fundamentally incompatible with the probabilistic interpretation of modern quantum mechanics. In this work, we present a pedagogical derivation of Bohr’s results directly from the time-independent Schrödinger equation in three dimensions. By applying two successive geometric constraints, first, a genuine dimensional reduction to a 2D planar system (not a naive restriction of the spherical Laplacian), and second, a radial localization to a fixed radius, we demonstrate that the resulting radial-azimuthal equation yields the quantized Bohr orbits. We show that the energy levels En = −13.6/n2 eV and orbital radii r[n] = n² ao emerge naturally from the wavefunction normalization combined with the de Broglie standing-wave condition. Special attention is given to the correct form of the 2D Laplacian, which differs from the 3D Laplacian restricted to θ = π/2 by a term (1/r) ∂/∂r. This approach provides a clear conceptual bridge between the diffuse quantum cloud and the classical planetary picture.

•Jun 8, 2026•0 citations

Derivations

R.C. Olsen

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.

SPIE eBooks•Mar 16, 2010•0 citations

Bohr Model

James W. Rohlf

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.

CRC Press eBooks•Oct 18, 2023•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

of Bits and Bonds: Hi-Tech Methods Predict the Newest Molecules: Part 2. Quantization

Anne‐Frances Miller

A quick tour through the development of Bohr’s model of the atom concludes with calculation of the allowable (quantized) energies of an electron orbiting in a Hydrogen atom. Electron states, and transitions between states are presented, as are their related spectra.

•Jan 1, 2013•0 citations

The Hydrogen-Like Atom

Uri Peskin

The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.

Cambridge University Press eBooks•May 11, 2023•0 citations
The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.
The Bohr model of the hydrogen atom, despite its heuristic success in predicting discrete energy levels, is fundamentally incompatible with the probabilistic interpretation of modern quantum mechanics. In this work, we present a pedagogical derivation of Bohr’s results directly from the time-independent Schrödinger equation in three dimensions. By applying two successive geometric constraints, first, a genuine dimensional reduction to a 2D planar system (not a naive restriction of the spherical Laplacian), and second, a radial localization to a fixed radius, we demonstrate that the resulting radial-azimuthal equation yields the quantized Bohr orbits. We show that the energy levels En = −13.6/n2 eV and orbital radii r[n] = n² ao emerge naturally from the wavefunction normalization combined with the de Broglie standing-wave condition. Special attention is given to the correct form of the 2D Laplacian, which differs from the 3D Laplacian restricted to θ = π/2 by a term (1/r) ∂/∂r. This approach provides a clear conceptual bridge between the diffuse quantum cloud and the classical planetary picture.
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.
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 mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.