Why is a triplet state said to have anti-parallel spins?

[Aniket1]

The triplet state can have one of the following three spin eigenfunctions:
(+1/2,+1/2)
(-1/2,-1/2)
(1/squareroot(2))*((+1/2,-1/2)+(-1/2,+1/2))
I read that this state has electrons with parallel spins (which i assume is the z component of the spin angular momentum)
I didn't understand how the third eigenfunction represents electrons with antiparallel spins when it assigns +1/2 to one electron and -1/2 to the other.

 

[daveyrocket]

The triplet state can either have total z-projected momentum of 1,0,-1. So when the electrons have parallel spins that can be state 1 or -1. When they have anti-parallel spins that is state 0. +1/2 and -1/2 are spins that point in the opposite direction, so that is the definition of anti-parallel (for a QM spin where you only have one quantization axis).

 

[Aniket1]

If +1/2 and -1/2 are spins pointing in opposite directions, the third spin eigenfunction (which has been mentioned in the question) should represent electrons with opposite spins ( or am i making a mistake in interpretating the function?). But it is an eigenfunction of the triplet state. And triplet states have parallel spins. So where is the flaw in the argument?
Also since the total z-projected momentum can be zero for a triplet state, the spins of the electrons in this case will be in opposite directions to cancel each other.
So what do we mean by parallel and antiparallel spins in triplet and singlet states?

 

[daveyrocket]

No triplet states don't just have parallel spins. The triplet state with m_z = 0 has antiparallel spins. But a pair of parallel spins must be a triplet state because it cannot be a singlet state. The singlet state has only antiparallel spins. The distinction between the singlet state | 0 0 > and the triplet state |1 0> is the sign the two states mix with.

 

[Aniket1]

Thanks..