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Fix Matter see-also invoke

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Fix Matter see-also invoke

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{{Short description|Discrete energy state of an electron in an atom}}

{{Quantum matter backlink|Atoms}}

An '''energy level''' is a discrete value of energy that an [[Physics:Quantum atoms/electron|electron]] can have in an [[Physics:Quantum atoms/atom|atom]]. These levels arise from the quantization of the electron’s motion and are one of the fundamental concepts of [[Physics:Quantum mechanics|quantum mechanics]].

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:470px;">
[[File:Atomic_energy_levels.png|450px]]
<div style="font-size:90%;">Discrete energy levels in an atom; transitions between them produce spectral lines.</div>
</div>

== Description ==
A quantum mechanical system that is spatially confined can only possess certain discrete values of energy, known as '''energy levels'''. This differs from classical systems, where energy may vary continuously. The concept is especially important for electrons bound to atomic nuclei, but also applies to molecular vibrations, rotations, and nuclear states.

Electrons in atoms can only occupy specific energy levels. These levels are determined by solving the [[Physics:Quantum mechanics/Schrödinger equation|Schrödinger equation]] for electrons bound to a nucleus. The allowed states correspond to standing-wave solutions of the electron wavefunction.<ref name="Tipler">{{cite book
| last1 = Tipler
| first1 = Paul A.
| last2 = Mosca
| first2 = Gene
| title = Physics for Scientists and Engineers, 5th Ed.
| publisher = W. H. Freeman and Co.
| volume = 2
| date = 2004
| pages = 1129
| url = https://books.google.com/books?id=R2Nuh3Ux1AwC&dq=%22energy+level%22+%22standing+waves%22&pg=PA1129
| isbn = 0716708108
}}</ref>

When an electron moves between energy levels it must absorb or emit energy, usually in the form of a photon. These transitions produce atomic spectral lines and form the basis of spectroscopy.

== Atomic energy levels ==
[[File:Energy levels.svg|thumb|right|Energy levels for an electron in an atom: ground state and excited states.]]

In atoms, the allowed electron energies are associated with shells and orbitals. The shells correspond to the principal quantum numbers {{math|''n'' = 1, 2, 3, ...}} and are often labeled K, L, M, and so forth.

Each shell can contain only a limited number of electrons. The maximum number is approximately given by:

:contentReference[oaicite:0]{index=0}

Thus the first shell holds two electrons, the second eight, and the third eighteen.<ref name="madsci">[http://www.madsci.org/posts/archives/1999-03/921736624.Ch.r.html Re: Why do electron shells have set limits ?] madsci.org, 17 March 1999, Dan Berger, Faculty Chemistry/Science, Bluffton College</ref>

The lowest possible energy configuration of an atom is called the '''ground state'''. Higher states are called '''excited states'''. An atom may contain several excited electrons simultaneously.

== Quantization ==
[[File:Hydrogen Density Plots.png|thumb|Wavefunctions of a hydrogen atom showing probability distributions for different energy levels.]]

Quantized energy levels arise because matter behaves as both particles and waves. Electrons confined near a nucleus form standing wave patterns. Only certain wavelengths satisfy the boundary conditions of the system, leading to discrete allowed energies.

For hydrogen-like atoms containing one electron, the energy levels are approximately given by:

:contentReference[oaicite:1]{index=1}

where {{math|''R''<sub>∞</sub>}} is the Rydberg constant, {{mvar|Z}} is the atomic number, and {{mvar|n}} is the principal quantum number.

The related Rydberg formula for emitted or absorbed wavelengths is:

:contentReference[oaicite:2]{index=2}

These relations successfully explain the spectral lines of hydrogen and hydrogen-like ions.

== Electron interactions ==
In multi-electron atoms, electron–electron interactions alter the simple hydrogenic energy levels. Inner electrons partially shield the positive nuclear charge, producing an effective nuclear charge {{math|''Z''<sub>eff</sub>}}.

An approximate expression becomes:

:contentReference[oaicite:3]{index=3}

These effects help determine electron configurations and atomic structure.

== Fine and hyperfine structure ==
Small corrections to energy levels arise from relativistic effects and spin interactions.

* '''Fine structure''' results from relativistic kinetic-energy corrections and spin–orbit coupling.
* '''Hyperfine structure''' results from interactions between electron spin and nuclear spin.

These small splittings are measurable with high-resolution spectroscopy.

== External field effects ==
=== Zeeman effect ===
External magnetic fields split atomic energy levels through interactions with magnetic moments associated with orbital and spin angular momentum.

The interaction energy is:

:contentReference[oaicite:4]{index=4}

This splitting produces the '''Zeeman effect'''.

=== Stark effect ===
External electric fields can also shift and split atomic energy levels, producing the '''Stark effect'''.

== Molecular energy levels ==
[[Chemical bond]]s in molecules also produce quantized energy states. Molecular systems possess:

* electronic energy levels
* vibrational energy levels
* rotational energy levels

The total molecular energy may be written as:

:contentReference[oaicite:5]{index=5}

Different molecular orbitals may be bonding, antibonding, or non-bonding.<ref name="chemguide">[http://www.chemguide.co.uk/analysis/uvvisible/theory.html#top UV-Visible Absorption Spectra]</ref>

== Energy level transitions ==
[[File:Atomic Absorption (hv corrected).svg|thumb|right|Absorption of a photon causing an electron transition to a higher energy level.]]

[[File:Schematic diagram of atomic line spontaneous emission (hv corrected).png|thumb|left|Emission of a photon during a transition to a lower energy level.]]

Electrons may transition between energy levels by absorbing or emitting photons. The photon energy equals the difference between the initial and final states:

:contentReference[oaicite:6]{index=6}

where {{math|''h''}} is the Planck constant, {{mvar|f}} is frequency, and {{mvar|λ}} is wavelength.

These transitions form the basis of atomic and molecular spectroscopy. Depending on the energy involved, transitions may emit or absorb radio waves, infrared radiation, visible light, ultraviolet radiation, or X-rays.

== History ==
The first evidence for quantized atomic energy levels came from the observation of spectral lines in sunlight by [[Biography:Joseph von Fraunhofer|Joseph von Fraunhofer]] and [[Biography:William Hyde Wollaston|William Hyde Wollaston]].

In 1913, [[Biography:Niels Bohr|Niels Bohr]] proposed quantized electron orbits in the Bohr model of the atom. Modern quantum mechanical explanations based on the Schrödinger equation were developed in the 1920s by [[Biography:Erwin Schrödinger|Erwin Schrödinger]] and [[Biography:Werner Heisenberg|Werner Heisenberg]].<ref name=Ruedenberg_Schwarz_2013>{{cite book | chapter=Three Millennia of Atoms and Molecules | first1=Klaus | last1=Ruedenberg | first2=W. H. Eugen | last2=Schwarz | title=Pioneers of Quantum Chemistry | pages=1–45 | series=ACS Symposium Series | volume=1122 | isbn=9780841227163 | date=February 13, 2013 | publisher=American Chemical Society | doi=10.1021/bk-2013-1122.ch001 }}</ref>

== Crystalline materials ==
In crystalline solids, electrons occupy energy bands rather than isolated levels. These bands arise because many closely spaced atomic energy levels overlap within the crystal lattice.

Important band-related quantities include:

* valence band
* conduction band
* Fermi level
* defect states

These determine the electrical and optical properties of materials.

== Properties ==

* quantized values
* associated with [[Physics:Quantum atoms/orbital|orbitals]]
* determine atomic spectra
* govern photon absorption and emission
* fundamental to atomic and molecular structure

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

=References=
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum atoms/energy level|1}}

Physics:Quantum atoms/energy level - Revision history

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imported>WikiHarold: Fix Matter see-also invoke