Triplet state

Two spin-1/2 particles

In a system with two spin-1/2 particlesfor example the proton and electron in the ground state of hydrogenmeasured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all

:\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

: |s_1,m_1\rangle|s_2,m_2\rangle = |s_1,m_1\rangle \otimes |s_2,m_2\rangle,

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

:|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

substituting in the four basis states

:\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}

returns the possible values for total spin given along with their representation in the \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:

: \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)}

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

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

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 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, spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on \R^3. Thus the "three" in triplet can be identified with the three rotation axes of physical space.

References

References

1. (2017). "Dioxygen: What Makes This Triplet Diradical Kinetically Persistent?". Journal of the American Chemical Society.
2. Townsend, John S.. (1992). "A modern approach to quantum mechanics". McGraw-Hill.
3. [https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_7.pdf Spin and Spin–Addition]