Product Operators: Effect of Rotations on Basis Operators of two spins (I=S=1/2)

Also: coupling converts : 2IxSy 2IxSy (no effect)

2IxSz 2IxSzcos? + Iysin?

2IySz 2IySzcos? - Ixsin?

What is the meaning of all this: I: magnetization spin 1, S: magnetization spin 2

Iz= not observable, equilibrium magnetization

Ix, Iy= observable magnetization of nuclei I (single quantum coherence)

IxSz, IySz = not observable (single quantum) antiphase magnetization antiphase, because it depends on spin state of other nucleus. The overall net magnetization of antiphase is 0, but it can evolve back into observable magnetization by coupling! Thus : if present after last pulse these terms will contribute to the observable signal.

IxSy = combination of zero and double quantum coherence, not observable; but can be converted back into single quantum coherence by a pulse. One can also separatethese terms into zero and double quantum coherences by using raising/lowering operators.

I+ = Ix + iIy Ix = 1/2 (I+ + I-) the coherence order is the

I- = Ix - iIy Iy = 1/2 (I+ - I-) sum of I + minus I- elements

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Note: Pulses excites the nuclei in a coherent way: they behave synchronously. After excitation they start precessing from the same point, all have the same frequency/ and move and behave identically. Thus, our excess magnetization will stay in sync and add up to a large signal. This is (commonly) called coherence. A special type of coherence are single quantum coherences ( in which only one type of nucleus is excited), if they are in the observable plane. We are able to detect those (and only those) with our receiver coil, and can commonly call them (observable) magnetization.

Try out: a) derive a product operator table for 90 degree pulses.

b) an example for precession: as in the vector presentation the two terms present the projections to the axes.

c) example for coupling= the cos- component is the projection to the starting axis, it does not precess. The other component consist of two vectors of opposite phase all the time.

d) go through precession of IxSy = four results= IxSycosIcosS + IxSxcosIsinS -IySysinIcosS-IySxsinIcosI