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Inquantum mechanics, atriplet state, orspin triplet, is thequantum state of an object such as an electron, atom, or molecule, having aquantum spinS = 1. It has three allowed values of the spin's projection along a given axism_S = −1, 0, or +1, giving the name "triplet".

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 individualatoms,protons, orelectrons.

A triplet state occurs in cases where the spins of twounpaired electrons, each having spins = 1/2, align to giveS = 1, in contrast to the more common case of two electrons aligning oppositely to giveS = 0, aspin singlet. Most molecules encountered in daily life exist in a singlet state because all of their electrons are paired, butmolecular oxygen is an exception.^[1] Atroom temperature, O₂ exists in a triplet state, which can only undergo a chemical reaction by making theforbidden transition into a singlet state. This makes it kinetically nonreactive despite being thermodynamically one of the strongest oxidants.Photochemical or thermalactivation can bring it into thesinglet state, which makes it kinetically as well as thermodynamically a very strong oxidant.

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

${\displaystyle \uparrow \uparrow ,\uparrow \downarrow ,\downarrow \uparrow ,\downarrow \downarrow }$

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

${\displaystyle |s_1,m_1⟩|s_2,m_2⟩=|s_1,m_1⟩\otimes |s_2,m_2⟩,}$

where${\displaystyle s_1}$ and${\displaystyle s_2}$ are the spins of the two particles, and${\displaystyle m_1}$ and${\displaystyle m_2}$ are their projections onto the z axis. Since for spin-1/2 particles, the$|\frac{1}{2},m⟩$ basis states span a 2-dimensional space, the$|\frac{1}{2},m_1⟩|\frac{1}{2},m_2⟩$ 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 inquantum mechanics using theClebsch–Gordan coefficients. In general

${\displaystyle |s,m⟩=\sum _{m_1+m_2=m}{C}_{m_1{m}_{2}m}^{s_1{s}_{2}s}|s_1{m}_1⟩|s_2{m}_2⟩}$

substituting in the four basis states

returns the possible values for total spin given along with their representation in the$|\frac{1}{2},m_1⟩|\frac{1}{2},m_2⟩$ basis. There are three states with total spin angular momentum 1:^[2][3]

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

${\displaystyle |0,0⟩=\frac{1}{\sqrt{2}}(\uparrow \downarrow -\downarrow \uparrow )\}s=0(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{t})}$

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.

In terms ofrepresentation 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-dimensionalClifford algebra) havetensored 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, ascalar, spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on${\displaystyle R^3}$. Thus the "three" in triplet can be identified with the three rotation axes of physical space.

• Griffiths, David J. (2004).Introduction to Quantum Mechanics (2nd ed.).Prentice Hall.ISBN 978-0-13-111892-8.
• Shankar, R. (1994). "chapter 14-Spin".Principles of Quantum Mechanics (2nd ed.). Springer.ISBN 978-0-306-44790-7.

• Quantum states
• Rotational symmetry
• Spectroscopy

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