Exploring NMR Operators: I_x, I_y, I_z

[Kreizhn]

I'm reading a paper on NMR, and the authors keep referring to the operators $I_x, I_y, I_z$. What are these operators? I keep finding them mentioned in other papers, but no description of what they are.

 

[Kreizhn]

In particular, these operators pertain to the discussion of coherent in-phase heteronuclear spin transfers. In this papers the operators mentioned are $S_x, S_y, S_z, I_x, I_y, I_z$. Am I correct in that these simply represent the spin operators for each particle?

 

[alxm]

The Pauli spin matrices
sigma_x = I_x = 1/2[0 1; 1 0], and so forth.

Which form the basis set for a 1/2 spin system with the |alpha> eigenket being [1;0] and |beta> being [0;1]

And to answer the second question: Yes.

 

[whybother]

I don't know specifically about NMR, but in quantum mechanics that I've come across, when S is spin, I are just the identity operators... but that would probably be obvious from the context.

 

[Kreizhn]

See, but the $S_x, S_y, S_z$ should also represent the spin operators. And in what sense would $I_x, I_y, I_z$ be three separate identity operators?

 

[Kreizhn]

What do you mean we have a different space for x,y,z of spin? A spin operator is a representation of actions of the special unitary group with appropriate dimension for the spin of the particle, acting on the same Hilbert space.

 

[alxm]

Ix,Iy,Iz is the notation used for spin operators in NMR, and Sx, Sy, Sz are the operators on the second spin when you're studying a two-spin system ("I-S").

Trust me. It's been a while since I studied NMR, but I do remember this much. Check out any book on the topic. Or google for some lecture notes or smth.

Edit: http://www.nmrfam.wisc.edu/~milo/notes/paradigm_II.pdf"

 

[Kreizhn]

So then you're confirming my second post?

 

[alxm]

So then you're confirming my second post?

Yup.

 

[Kreizhn]

Okay, the next thing is that given two heteronuclear spin 1/2 particles, the paper is considering the coherence-order selective in-phase transfer from $I^- \to S^-$ where $I^- = I_x - iI_y, S^- = S_x - iS_y$. But if spin I and spin S have the same representation, what does this transfer amount to? That is, could a concrete example be given for representations of I,S such that this is a nontrivial control problem?

 

[Kreizhn]

Nevermind, I think I found a book that's telling me that S and I are operators as considered on the combined spin system. This clears up my ambiguity.