# Exploring NMR Operators: I_x, I_y, I_z

• Kreizhn

#### 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.

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?

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.

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.

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?

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.

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.

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?