# How have I to understand in what those quantum numbers refer for ortho and para water?

I need to make an English presentation at my School (Graduate school) about something people may don't know. I chose to speak about the separation of ortho and para water. I first chat here with @Orthocresol about questions I have but this is really tricky and I'm not understanding that so much. So I chose to ask the question here, it might be helpful for other people and I hope at least for me.

At first my question is to understand what the notations $\lvert0_{00}0 \rangle$ and $\lvert1_{01}M \rangle$ in the article mention before about different states of water refers too?

What are those four numbers if the ket notation, from where they come from and what does they mean? I know it makes a lot of questions but maybe are they linked?

I read this article on wikipedia and this one too, and I also found that but I don't understood a lot of things. Maybe that $J$ is a quantum number which quantify the rotation energy of a molecule. But I have a lot of troubles with the projection and I don't know what $M$ is.

I would be glad if the answer was in a part like popularisation because I guess I'll have some troubles to understand it. But I have good skills in algebra, I had a twenty hours introduction in quantum chemistry, the hardest thing I did was a presentation of the spin-orbital coupling if it can help you to jugde my level on the subject. Thank you in advance for your answer.

## 1 Answer

It's explained in the first link you gave in the previous paragraph:

"The rotational quantum states of the asymmetric rotor, i. e., water, can be classified by $J_{K_{a}K_{c}}M$, with the total angular momentum quantum number $J$, the projection labels $K_{a}$ and $K_{c}$ onto the molecule-fixed a and c axes as defined in Figure 1 respectively, and the projection quantum number $M$ onto the space-fixed $Z$ axis."

Do you know how angular momentum in quantum mechanics works? You can look at the projection of the angular momentum onto a specific axis and the total angular momentum. Turns out the Heisenberg Uncertainty Principle prevents you from knowing the angular momentum projections along more than one axis at once...

• Why need I these representation of the states? Why only J is not enough, why to add its projections? And what M is .. ? Are we speaking about only nuclear spin or might it be also electron spin? – Hexacoordinate-C Nov 25 '16 at 2:12
• I think it's the molecular rotation. Otherwise, I don't think they would refer to it as an asymmetric rotor. Rotation about the a and c axes are such that the molecule is not symmetric about the axis. It is about the b axis though. Honestly, this is beyond the QM I know, and I would be hard pressed to explain it in clearer detail. – Zhe Nov 25 '16 at 2:59