Triplet State - Two Spin-1/2 Particles

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

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

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

and since for spin-1/2 particles, the basis states span a 2-dimensional space, the 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

substituting in the four basis states

returns the possible values for total spin given along with their representation in the basis. There are three states with total spin angular momentum 1

\left.\begin{align} |1,1\rangle &=\; \uparrow\uparrow\\ |1,0\rangle &=\; (\uparrow\downarrow + \downarrow\uparrow)/\sqrt2\\ |1,-1\rangle &=\; \downarrow\downarrow \end{align}\;\right\}\quad s=1\quad\mathrm{(triplet)}

and a fourth with total spin angular momentum 0.

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.

Read more about this topic:  Triplet State

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