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
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
where  and  are the spins of the two particles, and  and  are their projections onto the z axis.  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:
which are symmetric and a fourth state with total spin angular momentum 0:
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.