Editing Triplet state

{{Short description|Quantum state of a system}}
{{more footnotes needed|date=December 2010}}
[[File:Spin multiplicity diagram.svg|thumb|Examples of atoms in [[singlet state|singlet]], [[doublet state|doublet]], and '''triplet''' states.]]
In [[quantum mechanics]], a '''triplet state''', or '''spin triplet''', is the [[quantum state]] of an object such as an electron, atom, or molecule, having a [[Spin (physics)|quantum spin]] ''S'' = 1. It has three allowed values of the spin's projection along a given axis ''m''<sub>S</sub> = −1, 0, or +1, giving the name "triplet".

[[Spin (physics)|Spin]], in the context of quantum mechanics, is not a mechanical rotation but a more abstract concept that characterizes a particle's intrinsic angular momentum. It is particularly important for systems at atomic length scales, such as individual [[atoms]], [[protons]], or [[electrons]].

A triplet state occurs in cases where the spins of two [[unpaired electron]]s, each having spin ''s'' = {{frac|2}}, align to give ''S'' = 1, in contrast to the more common case of two electrons aligning oppositely to give ''S'' = 0, a [[spin singlet]]. Most molecules encountered in daily life exist in a singlet state because all of their electrons are paired, but [[molecular oxygen]] is an exception.<ref>{{cite journal |last1=Borden |first1=Weston Thatcher |last2=Hoffmann |first2=Roald |last3=Stuyver |first3=Thijs |last4=Chen |first4=Bo |date=2017 |title=Dioxygen: What Makes This Triplet Diradical Kinetically Persistent? |journal=Journal of the American Chemical Society |volume=139|issue=26 |pages=9010–9018 |doi=10.1021/jacs.7b04232 |pmid=28613073 |doi-access=free }}</ref> At [[room temperature]], O<sub>2</sub> exists in a triplet state, which can only undergo a chemical reaction by making the [[forbidden transition]] into a singlet state. This makes it kinetically nonreactive despite being thermodynamically one of the strongest oxidants. [[Photochemistry|Photochemical]] or thermal [[Activation energy|activation]] can bring it into the [[Singlet oxygen|singlet state]], which makes it kinetically as well as thermodynamically a very strong oxidant.

== Two spin-1/2 particles ==
In a system with two spin-1/2 particles{{snd}}for example the proton and electron in the ground state of hydrogen{{snd}}measured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all

:<math>\uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow</math>

using the single particle spins to label the basis states, where the first arrow and second arrow in each combination indicate the spin direction of the first particle and second particle respectively.

More rigorously

:<math>
|s_1,m_1\rangle|s_2,m_2\rangle = |s_1,m_1\rangle \otimes |s_2,m_2\rangle,
</math>

where <math>s_1</math> and <math>s_2</math> are the spins of the two particles, and <math>m_1</math> and <math>m_2</math> are their projections onto the z axis.  Since for spin-1/2 particles, the <math display="inline">\left|\frac{1}{2},m\right\rangle</math> basis states span a 2-dimensional space, the <math display="inline">\left|\frac{1}{2},m_1\right\rangle\left|\frac{1}{2},m_2\right\rangle</math> basis states span a 4-dimensional space.

Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in [[quantum mechanics]] using the [[Clebsch–Gordan coefficients]]. In general

:<math>|s,m\rangle = \sum_{m_1+m_2=m} C_{m_1m_2m}^{s_1s_2s}|s_1 m_1\rangle|s_2 m_2\rangle</math>

substituting in the four basis states

:<math>\begin{align}
  \left|\frac{1}{2},+\frac{1}{2}\right\rangle\ \otimes \left|\frac{1}{2},+\frac{1}{2}\right\rangle\ &\text{ by } (\uparrow\uparrow), \\
  \left|\frac{1}{2},+\frac{1}{2}\right\rangle\ \otimes \left|\frac{1}{2},-\frac{1}{2}\right\rangle\ &\text{ by } (\uparrow\downarrow), \\
  \left|\frac{1}{2},-\frac{1}{2}\right\rangle\ \otimes \left|\frac{1}{2},+\frac{1}{2}\right\rangle\ &\text{ by } (\downarrow\uparrow), \\
  \left|\frac{1}{2},-\frac{1}{2}\right\rangle\ \otimes \left|\frac{1}{2},-\frac{1}{2}\right\rangle\ &\text{ by } (\downarrow\downarrow)\end{align}</math>

returns the possible values for total spin given along with their representation in the <math display="inline">\left|\frac{1}{2},m_1\right\rangle\left|\frac{1}{2},m_2\right\rangle</math> basis. There are three states with total spin angular momentum 1:<ref>{{Cite book|last=Townsend|first=John S.|url=https://www.worldcat.org/oclc/23650343|title=A modern approach to quantum mechanics|page=149|date=1992|publisher=McGraw-Hill|isbn=0-07-065119-1|location=New York|oclc=23650343}}</ref><ref>[https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf Spin and Spin–Addition]</ref>

:<math>
\left.\begin{array}{ll}
|1,1\rangle &=\; \uparrow\uparrow \\
|1,0\rangle &=\; \frac{1}{\sqrt{2}}(\uparrow\downarrow + \downarrow\uparrow) \\
|1,-1\rangle &=\; \downarrow\downarrow
\end{array}\right\}\quad s = 1\quad \mathrm{(triplet)}
</math>

which are symmetric and a fourth state with total spin angular momentum 0:

:<math>\left.|0,0\rangle = \frac{1}{\sqrt{2}}(\uparrow\downarrow - \downarrow\uparrow)\;\right\}\quad s=0\quad\mathrm{(singlet)}</math>

which is antisymmetric. The result is that a combination of two spin-1/2 particles can carry a total spin of 1 or 0, depending on whether they occupy a triplet or singlet state.

== A mathematical viewpoint ==
In terms of [[representation theory]], what has happened is that the two conjugate 2-dimensional spin representations of the spin group SU(2) = Spin(3) (as it sits inside the 3-dimensional [[Clifford algebra]]) have [[tensor|tensored]] to produce a 4-dimensional representation. The 4-dimensional representation descends to the usual orthogonal group SO(3) and so its objects are tensors, corresponding to the integrality of their spin.  The 4-dimensional representation decomposes into the sum of a one-dimensional trivial representation (singlet, a [[scalar (physics)|scalar]], spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on <math>R^3</math>. Thus the "three" in triplet can be identified with the three rotation axes of physical space.{{citation needed|date=May 2025}}

== See also ==
{{cols|colwidth=26em}}
* [[Singlet state]]
* [[Doublet state]]
* [[Diradical]]
* [[Angular momentum]]
* [[Pauli matrices]]
* [[Spin multiplicity]]
* [[Spin quantum number]]
* [[Spin-1/2]]
* [[Spin tensor]]
* [[Spinor]]
{{colend}}

==References==
{{Reflist}}
*{{cite book | author=Griffiths, David J.|title=Introduction to Quantum Mechanics|edition=2nd | publisher=[[Prentice Hall]] |date=2004 |isbn=978-0-13-111892-8}}
*{{cite book | author=Shankar, R. | title=Principles of Quantum Mechanics | edition=2nd | publisher=Springer| date=1994 |isbn=978-0-306-44790-7 |chapter=chapter 14-Spin}}

[[Category:Quantum states]]
[[Category:Rotational symmetry]]
[[Category:Spectroscopy]]