COSY : the product operator description
Notes:
Note: in this table we only derived the terms for spin I . We get the same result for spin S with the same treatment starting from Sz.
These four terms are observed for spin I in a 2D-COSY spectrum. The first line represents the diagonal peaks: It is magnetization of spin I is frequency labeled in both time domains by the spin I frequency, and is therefore observed on the diagonal (wI,wI). In the direct dimension, we have the two standard Ix cos - Iysin components of our precessing magnetization.
The second row is magnetization of I, which has been transferred by the second pulse (itself is a mixing period!) to spin S. Again, we do have the two components of the precessing magnetization, but the terms are frequency labeled by I in the first, and by S in the direct domain. They give rise to an off-diagonal or cross peak, with the frequency (wI, wS) .
While the actual sin/cos modulations are not important, as we just change the phasing(mixing of real and imaginary parts) to get adsorptive signals, in this experiment the diagonal- and cross peaks are differently modulated (diagonal by cos cos, cross peaks by cos sin ). This means, that we can either get adsorptive line shape for diagonal or cross peaks, but not for them together.