Physics:Quantum Selection rules

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Selection rules in quantum mechanics determine which transitions between quantum states are allowed or forbidden when an atom interacts with electromagnetic radiation. These rules arise from symmetry properties of the system and the structure of the interaction Hamiltonian.

In atomic spectroscopy, the dominant mechanism is the electric dipole (E1) transition, which governs most observed spectral lines.

Electric dipole (E1) selection rules for atomic transitions: Δℓ = ±1, Δm = 0, ±1, with Δn unrestricted. Forbidden transitions include Δℓ = 0 and Δm ≠ 0, ±1.

Electric dipole (E1) selection rules

For electric dipole transitions, the allowed changes in quantum numbers are:

• $\mathrm{\Delta }\ell =\pm 1$
• $\mathrm{\Delta }m=0,\pm 1$
• $\mathrm{\Delta }n$ unrestricted

These rules follow from evaluating the dipole transition matrix elements between quantum states.

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Physical origin

Selection rules arise from transition matrix elements of the form:

$⟨{\psi }_f|\mathbf{𝐫}|{\psi }_i⟩$

A transition is allowed only if this integral is non-zero. This condition is governed by symmetry principles such as parity and angular momentum conservation.

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Angular momentum considerations

The selection rules reflect conservation of angular momentum:

• A photon carries one unit of angular momentum
• Therefore:

 $\mathrm{\Delta }\ell =\pm 1$

The magnetic quantum number depends on photon polarization:

• $\mathrm{\Delta }m=0$ (linear polarization)
• $\mathrm{\Delta }m=\pm 1$ (circular polarization)

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Forbidden transitions

Transitions that violate the E1 selection rules are called 'forbidden transitions. Typical examples include:

• $\mathrm{\Delta }\ell =0$
• $\mathrm{\Delta }m\ne 0,\pm 1$

These transitions have vanishing electric dipole matrix elements and therefore very low probability.

However, they may occur via higher-order interactions:

• Magnetic dipole (M1) transitions
• Electric quadrupole (E2) transitions

These processes are much weaker but are important in astrophysical plasmas and precision spectroscopy.

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Spectroscopic consequences

Selection rules determine:

• Which spectral lines are observed
• The relative transition probabilities
• The polarization properties of emitted radiation

They are essential for interpreting atomic spectra and identifying elements in laboratory and astrophysical environments.

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Relation to hydrogen atom

In the hydrogen atom, selection rules explain the structure of spectral series such as:

• Lyman series ($n\to 1$)
• Balmer series ($n\to 2$)

Only transitions satisfying the E1 selection rules contribute significantly to observed spectra.

See also

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

Foundations



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

Conceptual and interpretations



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

Mathematical structure and systems



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

Atomic and spectroscopy



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

Wavefunctions and modes



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

Quantum dynamics and evolution



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

Measurement and information



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

Quantum information and computing



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

Quantum optics and experiments



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
73. Template:Quantum optics operators

Open quantum systems



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

Quantum field theory



Structural dependency map of quantum field theory.

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

Statistical mechanics and kinetic theory



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

Condensed matter and solid-state physics



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

Plasma and fusion physics



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.

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

Timeline



125. Physics:Quantum mechanics/Timeline
126. Physics:Quantum mechanics/Timeline/Pre-quantum era
127. Physics:Quantum mechanics/Timeline/Old quantum theory
128. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
129. Physics:Quantum mechanics/Timeline/Quantum field theory era
130. Physics:Quantum mechanics/Timeline/Quantum information era
131. Physics:Quantum mechanics/Timeline/Quantum technology era
132. Physics:Quantum mechanics/Timeline/Quiz/

Advanced and frontier topics



133. Physics:Quantum Supersymmetry
134. Physics:Quantum Black hole thermodynamics
135. Physics:Quantum Holographic principle
136. Physics:Quantum gravity
137. Physics:Quantum De Sitter invariant special relativity
138. Physics:Quantum Doubly special relativity

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References

Author: Harold Foppele