Dihydrogen Spin Isomers

[splitringtail]

I am kinda rusty at my quantum, but I cannot figure out the coupling b/t ortho and para hydrogen forms. My quantum book doesn't discuss it and I really confused with the notation and superficial treatment I see on Wikipedia and other sites.

http://www.york.ac.uk/res/sbd/parahydrogen/outline.html

I understand why ortho is a triplet state b/c they are three orientations of the total spin $$m_{I}={1,0,-1}.$$ Yet, what are these alpha's and beta's on the site? Is the wave functions... both spin up, both spin down, and a linear combination of the two?. Then, for the para form, there's no difference if it is spin up down or down up, so they just take a linear combination and it's a singlet state?

 

[alxm]

"Ortho" and "Para" are just archaic terms for "triplet" and "singlet" electronic states, respectively. I wouldn't use the term 'isomers'. In fact I don't think I've ever seen the term 'spin isomer' before this. It's an excited electronic state of the same molecule. (I explain https://www.physicsforums.com/showpost.php?p=2120686&postcount=2" why triplet/ortho hydrogen is higher in energy, which usually extends to triplet states in general) Alpha and beta spins is just another way of saying 'up' and 'down' spins, or +1/2 and -1/2.