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{{DISPLAYTITLE:Selection rules}}
{{Quantum book backlink|Atomic and spectroscopy}}
'''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.<ref>[https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Spectroscopy/Fundamentals_of_Spectroscopy/Selection_rules Selection rules – LibreTexts]</ref>

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

[[File:Quantum_selection_rules_E1_atomic_transitions.jpg|thumb|400px|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:

* <math>\Delta \ell = \pm 1</math>
* <math>\Delta m = 0, \pm 1</math>
* <math>\Delta n</math> unrestricted

These rules follow from evaluating the dipole transition matrix elements between quantum states.<ref>[https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Spectroscopy Electronic spectroscopy – LibreTexts]</ref>

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== Physical origin ==

Selection rules arise from transition matrix elements of the form:

<math>
\langle \psi_f | \mathbf{r} | \psi_i \rangle
</math>

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.<ref>[https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Spectroscopy/Fundamentals_of_Spectroscopy/Selection_rules Selection rules – LibreTexts]</ref>

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== Angular momentum considerations ==

The selection rules reflect conservation of angular momentum:

* A photon carries one unit of angular momentum
* Therefore:
<math>\Delta \ell = \pm 1</math>

The magnetic quantum number depends on photon polarization:

* <math>\Delta m = 0</math> (linear polarization)
* <math>\Delta m = \pm 1</math> (circular polarization)

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== Forbidden transitions ==

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

* <math>\Delta \ell = 0</math>
* <math>\Delta m \neq 0, \pm 1</math>

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.<ref>[https://physics.nist.gov/PhysRefData/ASD/lines_form.html Atomic Spectra Database – NIST]</ref>

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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.<ref>[https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm Hydrogen spectral data – NIST]</ref>

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== Relation to hydrogen atom ==

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

* Lyman series (<math>n \to 1</math>)
* Balmer series (<math>n \to 2</math>)

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

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Selection rules|1}}

Physics:Quantum Selection rules - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template