2D-COSY
Consider: two homonuclear spin 1/2, coupled :
spectrum (1D) (first order)
In the vector diagram the rotation and pulse response of individual spins can nicely be presented. For two coupled spins one can similarly show vectors, and even splitting due to J. However, the exchange of magnetization between the spins, i.e. the effect of spin I on spin S and vice versa), can not be displayed in a graphical vector presentation. Based on the vector model a COSY experiment would only lead to two observable peaks, the diagonal peaks of both spins. To explain the crosspeaks/correlation peaks between the two spin, a quantum mechanical description is needed!
Usually: one would define the system/ spin state by wave functions, expand the wave function into a complete set of orthonormal functions, and ask for the expectation value. One obtains a matrix of (time dependent) coefficients= density matrix. Perturbation of the ensemble expectation values for different operators representing the pulse or precession perturbation.= leads to changes in the matrix /Operator, I.e. describes how one spin behaves under the influence of the magnetic field, pulses, time delays. For two (more) spins one could do the same, and one would start the QM description of a two spin system being a linear combination of the one-spin wavefunctions. One can show that this leads to the resulting operators being the product of the one-spin operators (E, Ix, Iy, Iz; and E, Sx, Sy, Sz). One obtains 16 Product operators = rules which describe the QM-response of a twp spin system.