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Triplet state
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<h2 id="two-spin-12-particles">Two spin-1/2 particles</h2> <p>In a system with two spin-1/2 particles – for example the proton and electron in the ground state of hydrogen – measured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all </p> ↑↑ , ↑↓ , ↓↑ , ↓↓ {\displaystyle \uparrow \uparrow ,\uparrow \downarrow ,\downarrow \uparrow ,\downarrow \downarrow } <p>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. </p><p>More rigorously </p> | s 1 , m 1 ⟩ | s 2 , m 2 ⟩ = | s 1 , m 1 ⟩ ⊗ | s 2 , m 2 ⟩ , {\displaystyle |s_{1},m_{1}\rangle |s_{2},m_{2}\rangle =|s_{1},m_{1}\rangle \otimes |s_{2},m_{2}\rangle ,} <p>where s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} are the spins of the two particles, and m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} are their projections onto the z axis. Since for spin-1/2 particles, the | 1 2 , m ⟩ {\textstyle \left|{\frac {1}{2}},m\right\rangle } basis states span a 2-dimensional space, the | 1 2 , m 1 ⟩ | 1 2 , m 2 ⟩ {\textstyle \left|{\frac {1}{2}},m_{1}\right\rangle \left|{\frac {1}{2}},m_{2}\right\rangle } basis states span a 4-dimensional space. </p><p>Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> using the <a href="/facts/Clebsch%E2%80%93Gordan_coefficients/r063mruI">Clebsch–Gordan coefficients</a>. In general </p> | s , m ⟩ = ∑ m 1 + m 2 = m C m 1 m 2 m s 1 s 2 s | s 1 m 1 ⟩ | s 2 m 2 ⟩ {\displaystyle |s,m\rangle =\sum _{m_{1}+m_{2}=m}C_{m_{1}m_{2}m}^{s_{1}s_{2}s}|s_{1}m_{1}\rangle |s_{2}m_{2}\rangle } <p>substituting in the four basis states </p> | 1 2 , + 1 2 ⟩ ⊗ | 1 2 , + 1 2 ⟩ by ( ↑↑ ) , | 1 2 , + 1 2 ⟩ ⊗ | 1 2 , − 1 2 ⟩ by ( ↑↓ ) , | 1 2 , − 1 2 ⟩ ⊗ | 1 2 , + 1 2 ⟩ by ( ↓↑ ) , | 1 2 , − 1 2 ⟩ ⊗ | 1 2 , − 1 2 ⟩ by ( ↓↓ ) {\displaystyle {\begin{aligned}\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{aligned}}} <p>returns the possible values for total spin given along with their representation in the | 1 2 , m 1 ⟩ | 1 2 , m 2 ⟩ {\textstyle \left|{\frac {1}{2}},m_{1}\right\rangle \left|{\frac {1}{2}},m_{2}\right\rangle } basis. There are three states with total spin angular momentum 1:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> | 1 , 1 ⟩ = ↑↑ | 1 , 0 ⟩ = 1 2 ( ↑↓ + ↓↑ ) | 1 , − 1 ⟩ = ↓↓ } s = 1 ( t r i p l e t ) {\displaystyle \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)} } <p>which are symmetric and a fourth state with total spin angular momentum 0: </p> | 0 , 0 ⟩ = 1 2 ( ↑↓ − ↓↑ ) } s = 0 ( s i n g l e t ) {\displaystyle \left.|0,0\rangle ={\frac {1}{\sqrt {2}}}(\uparrow \downarrow -\downarrow \uparrow )\;\right\}\quad s=0\quad \mathrm {(singlet)} } <p>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. </p> <h2 id="a-mathematical-viewpoint">A mathematical viewpoint</h2> <p>In terms of <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, 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 <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>) have <a href="/facts/Tensor/pNjFo5tV">tensored</a> 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 <a href="/facts/Scalar_(physics)/05uvvP8N">scalar</a>, spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on R 3 {\displaystyle R^{3}} . Thus the "three" in triplet can be identified with the three rotation axes of physical space. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Singlet_state/CK39GJa4">Singlet state</a></li> <li><a href="/facts/Doublet_state/R5Lr3xtg">Doublet state</a></li> <li><a href="/facts/Diradical/gH1vrH9b">Diradical</a></li> <li><a href="/facts/Angular_momentum/7TuSVFxq">Angular momentum</a></li> <li><a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a></li> <li><a href="/facts/Spin_multiplicity/4Jvp7e8P">Spin multiplicity</a></li> <li><a href="/facts/Spin_quantum_number/5Nh2H8gr">Spin quantum number</a></li> <li><a href="/facts/Spin-1/2/3I929M4X">Spin-1/2</a></li> <li><a href="/facts/Spin_tensor/GJgogobh">Spin tensor</a></li> <li><a href="/facts/Spinor/sKFNnEwP">Spinor</a></li></ul> <ul><li>Griffiths, David J. (2004). <i>Introduction to Quantum Mechanics</i> (2nd ed.). <a href="/facts/Prentice_Hall/bynWAuat">Prentice Hall</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-111892-8.</li> <li>Shankar, R. (1994). "chapter 14-Spin". <i>Principles of Quantum Mechanics</i> (2nd ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-306-44790-7.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Borden, Weston Thatcher; Hoffmann, Roald; Stuyver, Thijs; Chen, Bo (2017). "Dioxygen: What Makes This Triplet Diradical Kinetically Persistent?". Journal of the American Chemical Society. 139 (26): 9010–9018. doi:10.1021/jacs.7b04232. PMID 28613073. <a href="https://doi.org/10.1021%2Fjacs.7b04232" target="_blank">https://doi.org/10.1021%2Fjacs.7b04232</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Townsend, John S. (1992). A modern approach to quantum mechanics. New York: McGraw-Hill. p. 149. ISBN 0-07-065119-1. OCLC 23650343. <a href="0-07-065119-1" target="_blank">0-07-065119-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Spin and Spin–Addition <a href="https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf" target="_blank">https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>