View source for Energy level splitting

{{more citations needed|date=December 2009}}
[[Image:Breit-rabi-Zeeman-en.svg|thumb|right|350px|Splitting of [[atomic orbital|atomic energy levels]] in [[rubidium]]]]
In [[quantum physics]], '''energy level splitting''' or a split in an energy level of a quantum system occurs when a perturbation changes the system. The [[perturbation theory (quantum mechanics)|perturbation]] changes the corresponding [[Hamiltonian (quantum mechanics)|Hamiltonian]] and the outcome is change in [[eigenvalues]]; several distinct [[energy level]]s emerge in place of the former degenerate (multi-[[quantum state|state]]) level. This may occur because of external [[field (physics)|fields]], [[quantum tunnelling]] between states, or other effects. The term is most commonly used in reference to the [[electron configuration]] in [[atom]]s or [[molecule]]s.

The simplest case of level splitting is a [[qubit|quantum system with two states]] whose unperturbed Hamiltonian is a [[diagonal matrix|diagonal]] operator: {{math|1=''Ĥ''<sub>0</sub> = ''E''<sub>0</sub> ''I''}}, where {{math|''I''}} is the {{math|2 × 2}} [[identity matrix]]. Eigenstates and eigenvalues (energy levels) of a [[perturbation theory|perturbed]]<!-- is it a generic mathematical meaning? not something quantum-specific, right? --> Hamiltonian
:<math>\hat H_\varepsilon = \hat H_0+\varepsilon\sigma_3 =\begin{pmatrix} E_0+\varepsilon & 0 \\ 0 & E_0-\varepsilon \end{pmatrix}</math>
will be:
: {{math|{{pipe}}0{{rangle}}}}: the {{math|''E''<sub>0</sub> + ''ε'' }} level, and
: {{math|{{pipe}}1{{rangle}}}}: the {{math|''E''<sub>0</sub> − ''ε'' }} level,
so this degenerate {{math|''E''<sub>0</sub>}}&nbsp;eigenvalue splits in two whenever {{math|''ε'' ≠ 0}}. Though, if a perturbed Hamiltonian is not diagonal for this [[quantum state]]s basis {{math|{{{pipe}}0{{rangle}}, {{pipe}}1{{rangle}}} }}, then Hamiltonian's eigenstates are [[quantum superposition|linear combinations]] of these two states.

For a physical implementation such as a [[electric charge|charged]] [[spin-½]] particle [[Larmor precession|in an external magnetic field]], the ''z''-axis of the coordinate system is required to be collinear with the [[magnetic field]] to obtain a Hamiltonian in the form above (the {{math|''σ''<sub>3</sub>}}&nbsp;[[Pauli matrix]] corresponds to ''z''-axis). These basis states, referred to as ''[[Spin (physics)|spin]]-up'' and ''spin-down'', are hence eigenvectors of the perturbed Hamiltonian, so this level splitting is both easy to demonstrate mathematically and intuitively evident.

But in cases where the choice of state basis is ''not'' determined by a coordinate system, and the perturbed Hamiltonian is ''not'' diagonal, a level splitting may appear counter-intuitive, as in examples from chemistry below.

==Examples==
In [[atomic physics]]:
* The [[Zeeman effect]]&nbsp;– the splitting of electronic levels in an atom because of an external [[magnetic field]].
* The [[Stark effect]]&nbsp;– splitting because of an external [[electric field]].

In [[physical chemistry]]:
* The [[Jahn–Teller effect]]&nbsp;– splitting of electronic levels in a molecule because breaking the symmetry lowers the energy when the degenerate orbitals are partially filled.
* [[Resonance (chemistry)]] leads to creation of [[delocalized electron]] states.<small> ({{harvnb|Feynman|1965|loc=chapter&nbsp;10, §&nbsp;4}})</small>
* [[Nitrogen inversion]] leads to level splitting in [[ammonia]]<small> ({{harvnb|Feynman|1965|loc=chapter&nbsp;8, §&nbsp;6}})</small>, which is used in an ammonia [[maser]].<small> ({{harvnb|Feynman|1965|loc=chapter&nbsp;9}})</small>

== References ==
<!-- This is a required part of referencing, DO NOT REMOVE. See [[:Category:Harv and Sfn no-target errors]]. -->
* {{cite book |ref={{harvid|Feynman|1965}} |last1=Feynman |first1=Richard P. |author-link=Richard Feynman |author2=Robert Leighton |author3=Matthew Sands |title=The Feynman Lectures on Physics |volume=III |url=https://feynmanlectures.caltech.edu/III_toc.html |publisher=Addison-Wesley |year=1965 |location=Massachusetts, USA |isbn=0-201-02118-8}}

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[[Category:Quantum mechanics]]