Triplet state

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

In physics, spin is the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. In quantum mechanics, spin is particularly important for systems at atomic length scales, such as individual atoms, protons, or electrons. Such particles and the spin of quantum mechanical systems ("particle spin") possesses several unusual or non-classical features, and for such systems, spin angular momentum cannot be associated with rotation but instead refers only to the presence of angular momentum. A spin triplet is a set of three quantum states of a system, each with total spin S = 1. The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one (in units of <math>\hbar</math>).

Two Spin 1/2 Particles

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

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

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

and since for spin-1/2 particles, the <math>|1/2,m\rangle</math> basis states span a 2-dimensional space. Therefore the <math>|1/2,m_1\rangle|1/2,m_2\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>|j,m\rangle = \sum_{m_1+m_2=m}C_{m_1m_2m}^{s_1s_2s}|s_1m_1\rangle|s_2m_2\rangle</math>

substituting in the four basis states

<math> |1/2,+1/2\rangle|1/2,+1/2\rangle\ (\uparrow\uparrow)</math>

<math> |1/2,+1/2\rangle|1/2,-1/2\rangle\ (\uparrow\downarrow)</math>

<math> |1/2,-1/2\rangle|1/2,+1/2\rangle\ (\downarrow\uparrow)</math>

<math> |1/2,-1/2\rangle|1/2,-1/2\rangle\ (\downarrow\downarrow)</math>

returns the possible values for total spin given along with their representation in the <math>|1/2\ m_1\rangle|1/2\ m_2\rangle</math> basis. There are three states with total angular momentum 1

<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)\ s=1\ (\mathrm{triplet}) </math>

and a fourth with total angular momentum 0.

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

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.

See also

• Diradical
• Angular momentum
• Spin quantum number

References

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

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