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Mediterranean Journal of Basic and Applied Sciences

The Fundament of the Mass: The Variant Mass of Electron at Atom

Giovanni Alcocer

Published Jan 1, 2020• 6 citations
Publisher

Abstract

Experimental results: Ionization energy of the electrons at the atom.Bound of the Diatomic Molecules.From Bohr to Schröndiger.Deduction of the radius formula with the analogy of the atom with the blackbody research from Planck and harmonic oscillators.Why the electrons orbit around the nucleus with specific radius for the case of electron as particle.Why the electron doesn´t radiate energy at the stationary levels as a wave and as a particle.Diffraction and Interference for the electron by using the Fourier Approach and Wave Properties

DOI
10.46382/mjbas.2020.4402

Related papers

X Rays and Electron Diffraction & Interference and Diffraction for the electron by using the Fourier Approach and Wave Properties, The Variant Mass of the Electron at the atom & The Variant Mass for an Accelerated Charged Particle (Maxwell Radiation)

Giovanni Alcocer

The X rays Diffraction Experiment covers: theory, interaction of photons with the matter, sources of x rays, x rays tube, radioactivity, x rays detector, x rays diffraction process, crystals, miller indices and Bravais lattices, Bragg reflection on crystal, symmetry elements, point group of symmetry, fourteen bravais lattices, crystalline systems bravais, 32 crystalline punctual groups, stereogram of the 32 crystalline punctual groups, brillouin zone and the cubic system lattice, real space (direct lattices) and the reciprocal space (reciprocal lattices), the Laue experiment of x rays diffraction on crystal, experimental set up, scintillators and photomultipliers.Besides, it is researched the Electron Diffraction and the Diffraction and Interference for the electron by using the Fourier Approach and Wave Properties, Diffraction: Heisenberg's Uncertainty Principle, Uncertainty Principle of Heisenberg by applying wave properties, Interference: Double Slit Experiment, The Copenhagen Interpretation, Pilot wave interpretation and why the electron doesn´t radiate energy as a particle.For other hand, Albert Einstein wrote in a research article: "Does the inertia of a body depend on its energy content?" (Ist die Trägheit eines Körpers von seimen Energienhalt abhängig?):"If a body emits energy E in the form of radiation, its mass decreases by E/c 2 ".The fact that the energy that leaves from the body is converted into radiation energy makes no difference, so the more general conclusion is reached that the mass of a body is a measure of the content of its energy...It is not impossible that with bodies whose content of energy is highly variable (for example radio salts) the theory can be successfully tested.If the theory corresponds to the fact, radiation conducts inertia between the bodies that emit and absorb it 1 ".This is true for any type of radiation emitted (gravitational or electromagnetic energy) which produces a decrease in the mass of the body.Respect to the gravitational energy emission, it is demonstrated by theory, experiment and result the discovery formula which describe exactly the variant mass of a particle which emits gravitational energy which was demonstrated by myself at the article: The Fundament of the Mass and Effects of the Gravitation on a Particle and Light in the mass, time, distance, velocity, frequency, wavelength: Variant Mass for a Particle which emits Gravitational Energy for a particle orbiting a large Planet or Sun and for a Binary Star and Variant Frequency for the Light passing close a Gravitational Field from a Massive Object (Sun).The results of the mass formula are of great relevance for Gravitational Interactions.It is in accordance with the classic result for the emission of the total gravitational energy (bond total energy) for a particle orbiting a large Planet or Sun and for a Binary Star.At the atom, the electron only radiates this energy when it jumps from one orbit to another orbit at the atom.It is in accordance with the experimental results from the spectral lines of the atom.The difference is that in a gravitational field the particle or a planet around the sun can take any position at the space and any radius.But, the electron at the atom only can take restricted positions which are explained by quantum mechanics, and the electrons don´t emit radiation when they orbit around the nucleus.At this article, it is demonstrated the development formula for the variant mass of the electron at the atom which describe exactly the variant mass of the electron (charged particle) at the atom which emits electromagnetic energy from one stationary level to other.The results of the formula are compared with the ionization energy emission for the electron at the atom and the bound energy for the diatomic molecules.The results are in agreement with high accuracy.Besides, Maxwell's theory shows that electromagnetic waves are radiated whenever charges accelerate as for example for the electrón.Then, this electromagnetic radiation (photons) produces a decrease in the mass of the electron which is given by the formula of the Variant Mass for an Accelerated Charged Particle which was demonstrated by me at this research.Therefore, an additional objective of this article is to demonstrate by theory, calculations and results the discovered formula which describe exactly the variant mass of an accelerated charged particle.This charged particle emits electromagnetic radiation which is called the Maxwell Radiation.Also, the article analyzes and establishes about the effects of the variant mass on the particle.In addition, the formula is tested with the electromagnetic radiation emitted for an electron when leaves from the atom.Finally, it is obtained the formula for the power energy emitted for an Accelerated Charged Particle.

Mediterranean Journal of Basic and Applied Sciences•Jan 1, 2021•1 citation

BOHR'S ORBIT RADIUS HAS NOTHING TO DO WITH REDUCED MASS

Zhengtang Guo

This article points out an error in a large number of wellknown textbooks of modern physics, atomic physics and quantum mechanics. We prove that replacing the electron mass by the reduced mass in Bohr's equation for the radius of the electron does not yield Bohr's orbit radius, but rather the electron (orbit particle)-proton (nucleus) separation. Bohr's orbit radius has nothing to do with reduced mass. In energy formula, it is indeed correct to use the reduced mass. The conclusion also refers to exotic atoms.

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PREFACE PART I: THE STRUCTURE OF MATTER 1. THE MICROSCOPIC WORLD: ATOMS AND MOLECULES 1.1 Development of the Atomic Theory: Relative Atomic Weights 1.2 Atomic Magnitudes 1.3 The Charge-to-Mass Ratio of the Electron: Thomson's Method 1.4 The Charge of the Electron: Millikan's Method 1.5 Mass Spectrometry 1.6 The Atomic Mass Scale and the Mole 1.7 The Periodic Table 2. ORIGINS OF THE QUANTUM THEORY OF MATTER 2.1 The Franck-Hertz Experiment 2.2 The Photoelectric Effect 2.3 X-rays and Matter 2.4 The Emission Spectra of Atoms 2.5 The Nuclear Atom 2.6 The Problem of Black-Body Radiation 2.7 The Concept of Action 2.8 The Harmonic Oscillator 2.9 Action Quantized: The Heat Capacity of Solids 2.10 Some Orders of Magnitude 2.11 Bohr's Model of the Atom Appendix 2A: Rutherford Scattering 3. MATTER WAVES IN SIMPLE SYSTEMS 3.1 The de Broglie Hypothesis 3.2 The Nature of Waves 3.3 Dispersion Relations and Wave Equations: The Free Particle 3.4 Operators 3.5 Eigenfunctions and Eigenvalues 3.6 The Particle in a One-Dimensional Box 3.7 The Indeterminacy or Uncertainty Principle 3.8 Expectation Values Summary of Postulates 3.9 Particles in Two- and Three-Dimensional Boxes 3.10 Particles in Circular Boxes 3.11 Particles in Spherical Boxes 3.12 The Rigid Rotor Appendix 3A: More on Circular Coordinates and the Circular Box 4. PARTICLES IN VARYING POTENTIAL FIELDS TRANSITIONS 4.1 Finite Potential Barriers 4.2 The Quantum Mechanical Harmonic Oscillator 4.3 The Hydrogen Atom 4.4 The Shapes of Orbitals 4.5 Transitions between Energy Levels 5. THE STRUCTURE OF ATOMS 5.1 Electron Spin Magnetic Phenomena 5.2 The Pauli Exclusion Principle the Aufbau Principle 5.3 Electronic Configurations of Atoms 5.4 Calculation of Atomic Structures 5.5 Atomic Structure and Periodic Behavior 5.6 Term Splitting and the Vector Model 5.7 Fine Structure and Spin-Orbit Interactions Appendix 5A: The Stern-Gerlach Experiment 6. THE CHEMICAL BOND IN THE SIMPLEST MOLECULES: H[2+ AND H[2 6.1 Bonding Forces between Atoms 6.2 The Simplest Molecule: The Hydrogen Molecule-Ion, H[2+ 6.3 H[2+: Molecular Orbitals and the LCAO Approximation 6.4 H[2+: Obtaining the Energy Curve 6.5 H[2+: Correlation of Orbitals Excited States 6.6 The H[2 Molecule: Simple MO Description 6.7 Symmetry Properties of Identical Particles 6.8 H[2: The Valence Bond Representation 6.9 H[2: Beyond the Simple MO and VB Approximations 6.10 H[2: Excited Electronic States Appendix 6A: Orthogonality Appendix 6B: Hermitian Operators 7. MORE ABOUT DIATOMIC MOLECULES 7.1 Vibrations of Diatomic Molecules 7.2 Rotations of Diatomic Molecules 7.3 Spectra of Diatomic Molecules 7.4 The Ionic Bond 7.5 Homonuclear Diatomic Molecules: Molecular Orbitals and Orbital Correlation 7.6 Homonuclear Diatomic Molecules: Aufbau Principle and Structure of First-Row Molecules 7.7 Introduction to Heteronuclear Diatomic Molecules: Electronegativity 7.8 Bonding in LiH: Crossing and Noncrossing Potential Curves 7.9 Other First-Row Diatomic Hydrides 7.10 Isoelectronic and Other Series Appendix 7A: Perturbation Theory 8. TRIATOMIC MOLECULES 8.1 Electronic Structure and Geometry in the Simplest Cases: H[3 and H[3+ 8.2 Dihydrides: Introduction to the Water Molecule 8.3 Hybrid Orbitals 8.4 Delocalized Orbitals in H[2O: The General MO Method 8.5 Bonding in More Complex Triatomic Molecules 8.6 Normal Coordinates and Modes of Vibration 8.7 A Solvable Example: The Vibrational Modes of CO[2 8.8 Transitions and Spectra of Polyatomic Molecules: Rotations and Vibrations 8.9 Transitions and Spectra of Polyatomic Molecules: Magnetic Transitions 8.10 Transitions and Spectra of Polyatomic Molecules: Electronic Transitions 9. LARGER POLYATOMIC MOLECULES 9.1 Small Molecules 9.2 Catenated Carbon Compounds Transferability 9.3 Other Extended Structures 9.4 Some Steric Effects 9.5 Complex Ions and Other Coordination Compounds: Simple Polyhedra 9.6 Chirality and Optical Rotation 9.7 Chiral and Other Complex Ions 9.8 Magnetic Properties of Complexes 9.9 Electronic Structure of Complexes Appendix 9A: Schmidt Orthogonalization 10. INTERMOLECULAR FORCES 10.1 Long-Range Forces: Interactions between Charge Distributions 10.2 Empirical Intermolecular Potentials 10.3 Weakly Associated Molecules 11. THE STRUCTURE OF SOLIDS 11.1 Some General Properties of Solids 11.2 Space Lattices and Crystal Symmetry 11.3 X-ray Diffraction from Crystals: The Bragg Model 11.4 The Laue Model 11.5 Determination of Crystal Structures 11.6 Techniques of Diffraction 11.7 Molecular Crystals 11.8 Structures of Ionic Crystals 11.9 Binding Energy of Ionic Crystals 11.10 Covalent Solids 11.11 The Free-Electron Theory of Metals 11.12 The Band Theory of Solids 11.13 Conductors, Insulators, and Semiconductors 11.14 Other Forms of Condensed Matter APPENDICES I. Systems of Units II. Partial Derivatives III. Glossary of Symbols IV. Searching the Scientific Literature INDEX

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Bohr's Atomic Model Revisited

Francisco Caruso, Vitor Oguri

Bohr's atomic model, its relationship to the radiation spectrum of the hydrogen atom and the inherent hypotheses are revisited. It is argued that Bohr could have adopted a different approach, focusing his analyzes on the stationary orbit of the electron and its decomposition on two harmonic oscillators and then imposing, as actually he did, Planck's quantization for the oscillators' energies. Some consequences of this procedure are examined.

arXiv:0806.0652•Old and New Concepts of Physics•Apr 1, 2009•1 citation

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
Experimental results: Ionization energy of the electrons at the atom.Bound of the Diatomic Molecules.From Bohr to Schröndiger.Deduction of the radius formula with the analogy of the atom with the blackbody research from Planck and harmonic oscillators.Why the electrons orbit around the nucleus with specific radius for the case of electron as particle.Why the electron doesn´t radiate energy at the stationary levels as a wave and as a particle.Diffraction and Interference for the electron by using the Fourier Approach and Wave Properties
The X rays Diffraction Experiment covers: theory, interaction of photons with the matter, sources of x rays, x rays tube, radioactivity, x rays detector, x rays diffraction process, crystals, miller indices and Bravais lattices, Bragg reflection on crystal, symmetry elements, point group of symmetry, fourteen bravais lattices, crystalline systems bravais, 32 crystalline punctual groups, stereogram of the 32 crystalline punctual groups, brillouin zone and the cubic system lattice, real space (direct lattices) and the reciprocal space (reciprocal lattices), the Laue experiment of x rays diffraction on crystal, experimental set up, scintillators and photomultipliers.Besides, it is researched the Electron Diffraction and the Diffraction and Interference for the electron by using the Fourier Approach and Wave Properties, Diffraction: Heisenberg's Uncertainty Principle, Uncertainty Principle of Heisenberg by applying wave properties, Interference: Double Slit Experiment, The Copenhagen Interpretation, Pilot wave interpretation and why the electron doesn´t radiate energy as a particle.For other hand, Albert Einstein wrote in a research article: "Does the inertia of a body depend on its energy content?" (Ist die Trägheit eines Körpers von seimen Energienhalt abhängig?):"If a body emits energy E in the form of radiation, its mass decreases by E/c 2 ".The fact that the energy that leaves from the body is converted into radiation energy makes no difference, so the more general conclusion is reached that the mass of a body is a measure of the content of its energy...It is not impossible that with bodies whose content of energy is highly variable (for example radio salts) the theory can be successfully tested.If the theory corresponds to the fact, radiation conducts inertia between the bodies that emit and absorb it 1 ".This is true for any type of radiation emitted (gravitational or electromagnetic energy) which produces a decrease in the mass of the body.Respect to the gravitational energy emission, it is demonstrated by theory, experiment and result the discovery formula which describe exactly the variant mass of a particle which emits gravitational energy which was demonstrated by myself at the article: The Fundament of the Mass and Effects of the Gravitation on a Particle and Light in the mass, time, distance, velocity, frequency, wavelength: Variant Mass for a Particle which emits Gravitational Energy for a particle orbiting a large Planet or Sun and for a Binary Star and Variant Frequency for the Light passing close a Gravitational Field from a Massive Object (Sun).The results of the mass formula are of great relevance for Gravitational Interactions.It is in accordance with the classic result for the emission of the total gravitational energy (bond total energy) for a particle orbiting a large Planet or Sun and for a Binary Star.At the atom, the electron only radiates this energy when it jumps from one orbit to another orbit at the atom.It is in accordance with the experimental results from the spectral lines of the atom.The difference is that in a gravitational field the particle or a planet around the sun can take any position at the space and any radius.But, the electron at the atom only can take restricted positions which are explained by quantum mechanics, and the electrons don´t emit radiation when they orbit around the nucleus.At this article, it is demonstrated the development formula for the variant mass of the electron at the atom which describe exactly the variant mass of the electron (charged particle) at the atom which emits electromagnetic energy from one stationary level to other.The results of the formula are compared with the ionization energy emission for the electron at the atom and the bound energy for the diatomic molecules.The results are in agreement with high accuracy.Besides, Maxwell's theory shows that electromagnetic waves are radiated whenever charges accelerate as for example for the electrón.Then, this electromagnetic radiation (photons) produces a decrease in the mass of the electron which is given by the formula of the Variant Mass for an Accelerated Charged Particle which was demonstrated by me at this research.Therefore, an additional objective of this article is to demonstrate by theory, calculations and results the discovered formula which describe exactly the variant mass of an accelerated charged particle.This charged particle emits electromagnetic radiation which is called the Maxwell Radiation.Also, the article analyzes and establishes about the effects of the variant mass on the particle.In addition, the formula is tested with the electromagnetic radiation emitted for an electron when leaves from the atom.Finally, it is obtained the formula for the power energy emitted for an Accelerated Charged Particle.
This article points out an error in a large number of wellknown textbooks of modern physics, atomic physics and quantum mechanics. We prove that replacing the electron mass by the reduced mass in Bohr's equation for the radius of the electron does not yield Bohr's orbit radius, but rather the electron (orbit particle)-proton (nucleus) separation. Bohr's orbit radius has nothing to do with reduced mass. In energy formula, it is indeed correct to use the reduced mass. The conclusion also refers to exotic atoms.
PREFACE PART I: THE STRUCTURE OF MATTER 1. THE MICROSCOPIC WORLD: ATOMS AND MOLECULES 1.1 Development of the Atomic Theory: Relative Atomic Weights 1.2 Atomic Magnitudes 1.3 The Charge-to-Mass Ratio of the Electron: Thomson's Method 1.4 The Charge of the Electron: Millikan's Method 1.5 Mass Spectrometry 1.6 The Atomic Mass Scale and the Mole 1.7 The Periodic Table 2. ORIGINS OF THE QUANTUM THEORY OF MATTER 2.1 The Franck-Hertz Experiment 2.2 The Photoelectric Effect 2.3 X-rays and Matter 2.4 The Emission Spectra of Atoms 2.5 The Nuclear Atom 2.6 The Problem of Black-Body Radiation 2.7 The Concept of Action 2.8 The Harmonic Oscillator 2.9 Action Quantized: The Heat Capacity of Solids 2.10 Some Orders of Magnitude 2.11 Bohr's Model of the Atom Appendix 2A: Rutherford Scattering 3. MATTER WAVES IN SIMPLE SYSTEMS 3.1 The de Broglie Hypothesis 3.2 The Nature of Waves 3.3 Dispersion Relations and Wave Equations: The Free Particle 3.4 Operators 3.5 Eigenfunctions and Eigenvalues 3.6 The Particle in a One-Dimensional Box 3.7 The Indeterminacy or Uncertainty Principle 3.8 Expectation Values Summary of Postulates 3.9 Particles in Two- and Three-Dimensional Boxes 3.10 Particles in Circular Boxes 3.11 Particles in Spherical Boxes 3.12 The Rigid Rotor Appendix 3A: More on Circular Coordinates and the Circular Box 4. PARTICLES IN VARYING POTENTIAL FIELDS TRANSITIONS 4.1 Finite Potential Barriers 4.2 The Quantum Mechanical Harmonic Oscillator 4.3 The Hydrogen Atom 4.4 The Shapes of Orbitals 4.5 Transitions between Energy Levels 5. THE STRUCTURE OF ATOMS 5.1 Electron Spin Magnetic Phenomena 5.2 The Pauli Exclusion Principle the Aufbau Principle 5.3 Electronic Configurations of Atoms 5.4 Calculation of Atomic Structures 5.5 Atomic Structure and Periodic Behavior 5.6 Term Splitting and the Vector Model 5.7 Fine Structure and Spin-Orbit Interactions Appendix 5A: The Stern-Gerlach Experiment 6. THE CHEMICAL BOND IN THE SIMPLEST MOLECULES: H[2+ AND H[2 6.1 Bonding Forces between Atoms 6.2 The Simplest Molecule: The Hydrogen Molecule-Ion, H[2+ 6.3 H[2+: Molecular Orbitals and the LCAO Approximation 6.4 H[2+: Obtaining the Energy Curve 6.5 H[2+: Correlation of Orbitals Excited States 6.6 The H[2 Molecule: Simple MO Description 6.7 Symmetry Properties of Identical Particles 6.8 H[2: The Valence Bond Representation 6.9 H[2: Beyond the Simple MO and VB Approximations 6.10 H[2: Excited Electronic States Appendix 6A: Orthogonality Appendix 6B: Hermitian Operators 7. MORE ABOUT DIATOMIC MOLECULES 7.1 Vibrations of Diatomic Molecules 7.2 Rotations of Diatomic Molecules 7.3 Spectra of Diatomic Molecules 7.4 The Ionic Bond 7.5 Homonuclear Diatomic Molecules: Molecular Orbitals and Orbital Correlation 7.6 Homonuclear Diatomic Molecules: Aufbau Principle and Structure of First-Row Molecules 7.7 Introduction to Heteronuclear Diatomic Molecules: Electronegativity 7.8 Bonding in LiH: Crossing and Noncrossing Potential Curves 7.9 Other First-Row Diatomic Hydrides 7.10 Isoelectronic and Other Series Appendix 7A: Perturbation Theory 8. TRIATOMIC MOLECULES 8.1 Electronic Structure and Geometry in the Simplest Cases: H[3 and H[3+ 8.2 Dihydrides: Introduction to the Water Molecule 8.3 Hybrid Orbitals 8.4 Delocalized Orbitals in H[2O: The General MO Method 8.5 Bonding in More Complex Triatomic Molecules 8.6 Normal Coordinates and Modes of Vibration 8.7 A Solvable Example: The Vibrational Modes of CO[2 8.8 Transitions and Spectra of Polyatomic Molecules: Rotations and Vibrations 8.9 Transitions and Spectra of Polyatomic Molecules: Magnetic Transitions 8.10 Transitions and Spectra of Polyatomic Molecules: Electronic Transitions 9. LARGER POLYATOMIC MOLECULES 9.1 Small Molecules 9.2 Catenated Carbon Compounds Transferability 9.3 Other Extended Structures 9.4 Some Steric Effects 9.5 Complex Ions and Other Coordination Compounds: Simple Polyhedra 9.6 Chirality and Optical Rotation 9.7 Chiral and Other Complex Ions 9.8 Magnetic Properties of Complexes 9.9 Electronic Structure of Complexes Appendix 9A: Schmidt Orthogonalization 10. INTERMOLECULAR FORCES 10.1 Long-Range Forces: Interactions between Charge Distributions 10.2 Empirical Intermolecular Potentials 10.3 Weakly Associated Molecules 11. THE STRUCTURE OF SOLIDS 11.1 Some General Properties of Solids 11.2 Space Lattices and Crystal Symmetry 11.3 X-ray Diffraction from Crystals: The Bragg Model 11.4 The Laue Model 11.5 Determination of Crystal Structures 11.6 Techniques of Diffraction 11.7 Molecular Crystals 11.8 Structures of Ionic Crystals 11.9 Binding Energy of Ionic Crystals 11.10 Covalent Solids 11.11 The Free-Electron Theory of Metals 11.12 The Band Theory of Solids 11.13 Conductors, Insulators, and Semiconductors 11.14 Other Forms of Condensed Matter APPENDICES I. Systems of Units II. Partial Derivatives III. Glossary of Symbols IV. Searching the Scientific Literature INDEX
In this work, we reanalyzed the movement of an electron in the electrostatic field of nucleus. The trajectory of the electron’s motion is an ellipse with a minor semiaxis, tending towards zero. From a mathematical point of view the movement of an electron in such an orbit will be equivalent to the oscillation of an electron. The action produced by electrons in movement between stationary points is discrete and proportional to a Planck constant. This condition sets the allowable values of the electron energy and the radius of their orbit. Electrons on the same shell perform symmetric synchronous oscillations. Their frequency is of the order of 1016 Hz. Most of the time the electrons are located on the periphery of the atom, periodically they simultaneously rush to the nucleus, the atom rapidly compresses and immediately decompresses, i.e. pulsates. The model gives Bohr formula for the energy of single-electron atom and suitable values of ionization potentials of the atoms of the second period of the Periodic Table.
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.