Physics:Quantum Hydrogen atom

← Back to Atomic and spectroscopy

The hydrogen atom is the simplest atomic system, consisting of a single electron bound to a proton by the Coulomb interaction. It is the only atom in quantum mechanics that admits a fully exact analytical solution of the Schrödinger equation, making it a fundamental model for understanding atomic structure, spectroscopy, and quantum theory.

Energy levels and spectral series of the hydrogen atom showing Lyman (ultraviolet), Balmer (visible), and Paschen, Brackett, Pfund (infrared) transitions.

Schrödinger equation and Coulomb potential

The electron in a hydrogen atom is described by the time-independent Schrödinger equation in a central Coulomb potential:

$[-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2-\frac{e^2}{4\pi {\epsilon }_{0}r}]\psi (\mathbf{𝐫})=E\psi (\mathbf{𝐫})$

Because the potential depends only on the radial coordinate $r$, the equation is separable in spherical coordinates.

---

Quantum numbers

The solutions are characterized by three quantum numbers:

• Principal quantum number: $n=1,2,3,\dots$
• Orbital angular momentum: $\ell =0,1,\dots ,n-1$
• Magnetic quantum number: $m=-\ell ,\dots ,\ell$

These arise from the separation of variables into radial and angular parts.

---

Energy levels

The allowed energy levels depend only on the principal quantum number:

$E_n=-\frac{13.6\text{eV}}{n^2}$

This degeneracy is a consequence of the underlying symmetry of the Coulomb potential.

---

Wavefunctions and orbitals

The hydrogen wavefunctions are products of radial functions and spherical harmonics:

${\psi }_{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\varphi )$

These define the familiar atomic orbitals:

• $s$-orbitals ($\ell =0$) — spherical symmetry
• $p$-orbitals ($\ell =1$) — directional lobes
• $d$-orbitals ($\ell =2$) — more complex structures

---

Angular momentum

The orbital angular momentum is quantized:

$L^2={\hslash }^2\ell (\ell +1),L_z=\hslash m$

The hydrogen atom also includes electron spin, introducing total angular momentum when relativistic effects are considered.

---

Spectral lines

Transitions between energy levels produce photons with energy:

$E=h\nu =E_i-E_f$

This gives rise to discrete spectral series:

• Lyman series ($n\to 1$) — ultraviolet
• Balmer series ($n\to 2$) — visible
• Paschen, Brackett, Pfund — infrared

The wavelengths satisfy the Rydberg formula:

$\frac{1}{\lambda }=R_H(\frac{1}{n_f^2}-\frac{1}{n_i^2})$

where $R_H$ is the Rydberg constant.

---

Fine and hyperfine structure

More accurate treatments include:

• Fine structure — relativistic corrections and spin–orbit coupling
• Hyperfine structure — interaction between electron and nuclear spin

These effects lift degeneracies and produce small spectral splittings.

---

Importance in quantum mechanics

The hydrogen atom plays a central role because:

• It provides an exact solution of the Schrödinger equation
• It explains atomic spectra quantitatively
• It reveals hidden symmetries (e.g., Runge–Lenz vector)
• It serves as the starting point for multi-electron approximations

---

Table of contents (136 articles)

Index

Core theory

1. Foundations
2. Conceptual and interpretations
3. Mathematical structure and systems
4. Atomic and spectroscopy
5. Wavefunctions and modes
6. Quantum dynamics and evolution
7. Measurement and information
8. Quantum information and computing

Matter (by scale)

9. Matter (by scale)

Applications and extensions

10. Quantum optics and experiments
11. Open quantum systems
12. Quantum field theory
13. Statistical mechanics and kinetic theory
14. Condensed matter and solid-state physics
15. Plasma and fusion physics
16. Timeline
17. Advanced and frontier topics

Methods and tools

18. Methods and tools

Full contents

1. Foundations (7)↑

1. Physics:Quantum basics
2. Physics:Quantum Postulates
3. Physics:Quantum Hilbert space
4. Physics:Quantum Observables and operators
5. Physics:Quantum mechanics
6. Physics:Quantum mechanics measurements
7. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (10)↑

8. Physics:Quantum Interpretations of quantum mechanics
9. Physics:Quantum Wave–particle duality
10. Physics:Quantum Complementarity principle
11. Physics:Quantum Uncertainty principle
12. Physics:Quantum Measurement problem
13. Physics:Quantum Bell's theorem
14. Physics:Quantum Hidden variable theory
15. Physics:Quantum A Spooky Action at a Distance
16. Physics:Quantum A Walk Through the Universe
17. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

3. Mathematical structure and systems (11)↑

18. Physics:Quantum Density matrix
19. Physics:Quantum Exactly solvable quantum systems
20. Physics:Quantum Formulas Collection
21. Physics:Quantum A Matter Of Size
22. Physics:Quantum Symmetry in quantum mechanics
23. Physics:Quantum Angular momentum operator
24. Physics:Quantum Runge–Lenz vector
25. Physics:Quantum Approximation Methods
26. Physics:Quantum Matter Elements and Particles
27. Physics:Quantum Dirac equation
28. Physics:Quantum Klein–Gordon equation

4. Atomic and spectroscopy (11)↑

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

29. Physics:Quantum Atomic structure and spectroscopy
30. Physics:Quantum Hydrogen atom
31. Physics:Quantum Multi-electron atoms
32. Physics:Quantum Fine structure
33. Physics:Quantum Hyperfine structure
34. Physics:Quantum Isotopic shift
35. Physics:Quantum Zeeman effect
36. Physics:Quantum Stark effect
37. Physics:Quantum Spectral lines and series
38. Physics:Quantum Selection rules
39. Physics:Quantum Fermi's golden rule

5. Wavefunctions and modes (8)↑

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

40. Physics:Quantum Wavefunction
41. Physics:Quantum Superposition principle
42. Physics:Quantum Eigenstates and eigenvalues
43. Physics:Quantum Boundary conditions and quantization
44. Physics:Quantum Standing waves and modes
45. Physics:Quantum Normal modes and field quantization
46. Physics:Number of independent spatial modes in a spherical volume
47. Physics:Quantum Density of states

6. Quantum dynamics and evolution (9)↑

48. Physics:Quantum Time evolution
49. Physics:Quantum Schrödinger equation
50. Physics:Quantum Time-dependent Schrödinger equation
51. Physics:Quantum Stationary states
52. Physics:Quantum Perturbation theory
53. Physics:Quantum Time-dependent perturbation theory
54. Physics:Quantum Adiabatic theorem
55. Physics:Quantum Scattering theory
56. Physics:Quantum S-matrix

7. Measurement and information (6)↑

57. Physics:Quantum Measurement theory
58. Physics:Quantum Measurement operators
59. Physics:Quantum Projective measurement
60. Physics:Quantum POVM
61. Physics:Quantum Weak measurement
62. Physics:Quantum Measurement collapse

8. Quantum information and computing (6)↑

63. Physics:Quantum information theory
64. Physics:Quantum Qubit
65. Physics:Quantum Entanglement
66. Physics:Quantum Gates and circuits
67. Physics:Quantum Computing Algorithms in the NISQ Era
68. Physics:Quantum Noisy Qubits

9. Quantum optics and experiments (4)↑

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

69. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
70. Physics:Quantum optics beam splitter experiments
71. Physics:Quantum Ultra fast lasers
72. Physics:Quantum Experimental quantum physics
Template:Quantum optics operators

10. Open quantum systems (7)↑

73. Physics:Quantum Open systems
74. Physics:Quantum Master equation
75. Physics:Quantum Lindblad equation
76. Physics:Quantum Decoherence
77. Physics:Quantum Markovian dynamics
78. Physics:Quantum Non-Markovian dynamics
79. Physics:Quantum Trajectories

11. Quantum field theory (18)↑

Structural dependency map of quantum field theory.

80. Physics:Quantum field theory (QFT) basics
81. Physics:Quantum field theory (QFT) core
82. Physics:Quantum Fields and Particles
83. Physics:Quantum Second quantization
84. Physics:Quantum Harmonic Oscillator field modes
85. Physics:Quantum Creation and annihilation operators
86. Physics:Quantum vacuum fluctuations
87. Physics:Quantum Propagators in quantum field theory
88. Physics:Quantum Feynman diagrams
89. Physics:Quantum Path integral formulation
90. Physics:Quantum Renormalization in field theory
91. Physics:Quantum Renormalization group
92. Physics:Quantum Field Theory Gauge symmetry
93. Physics:Quantum Non-Abelian gauge theory
94. Physics:Quantum Electrodynamics (QED)
95. Physics:Quantum chromodynamics (QCD)
96. Physics:Quantum Electroweak theory
97. Physics:Quantum Standard Model

12. Statistical mechanics and kinetic theory (9)↑

98. Physics:Quantum Statistical mechanics
99. Physics:Quantum Partition function
100. Physics:Quantum Distribution functions
101. Physics:Quantum Liouville equation
102. Physics:Quantum Kinetic theory
103. Physics:Quantum Boltzmann equation
104. Physics:Quantum BBGKY hierarchy
105. Physics:Quantum Relaxation and thermalization
106. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (7)↑

107. Physics:Quantum Band structure
108. Physics:Quantum Fermi surfaces
109. Physics:Quantum Semiconductor physics
110. Physics:Quantum Phonons
111. Physics:Quantum Electron-phonon interaction
112. Physics:Quantum Superconductivity
113. Physics:Quantum Topological phases of matter

14. Plasma and fusion physics (8)↑

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

114. Physics:Quantum Fusion reactions and Lawson criterion
115. Physics:Quantum Plasma (fusion context)
116. Physics:Quantum Magnetic confinement fusion
117. Physics:Quantum Inertial confinement fusion
118. Physics:Quantum Plasma instabilities and turbulence
119. Physics:Quantum Tokamak core plasma
120. Physics:Quantum Tokamak edge physics and recycling asymmetries
121. Physics:Quantum Stellarator

15. Timeline (8)↑

122. Physics:Quantum mechanics/Timeline
123. Physics:Quantum mechanics/Timeline/Pre-quantum era
124. Physics:Quantum mechanics/Timeline/Old quantum theory
125. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
126. Physics:Quantum mechanics/Timeline/Quantum field theory era
127. Physics:Quantum mechanics/Timeline/Quantum information era
128. Physics:Quantum mechanics/Timeline/Quantum technology era
129. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (6)↑

130. Physics:Quantum Supersymmetry
131. Physics:Quantum Black hole thermodynamics
132. Physics:Quantum Holographic principle
133. Physics:Quantum gravity
134. Physics:Quantum De Sitter invariant special relativity
135. Physics:Quantum Doubly special relativity

References

Author: Harold Foppele