Editing Triplet state (section)

== 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.