spins. All observable(and non-observable) physical values can be extracted by multiplying the density matrix with their appropriate operator and then calculating the trace of the resulting matrix The time-dependent evolution of the system is calculated by unitary transformations(corresponding to"rotations")of the density matrix operator with the proper Hamiltonian H (including r f. pulses, chemical shift evolution, J coupling etc. p(t')=expiHt) p(t)exp(-iHt (for calculations these exponential operators have to be expanded into a Taylor series) The density operator can be written als linear combination of a set of basis operators. Two specific bases turn out to be useful for NMR experiments the real Cartesian product operators Ix, Iy and Iz(useful for description of observable magnetization and effects of r.f. pulses, J coupling and chemical shift)and the complex single-element basis set I, I, I and I(raising / lowering operators, useful for coherence order selection/ phase cycling/gradient selection) Cartesian Product operators Lit. O.W. Sorensen et al. (1983), Prog. NMR. Spectr: 16, 163-192 ngle spin operators correspond to magnetization of single spins, analogous to the classical macroscopic magnetization Mx, My, Mz (in-phase coherence, observable) ( polarisation, not observable Two-spin operators 2I1xl2:, 211y12=, 211z12x, 211212 (antiphase coherence, not observable) (longitudinal two-spin order, not observable) 211x12x, 2l1yl2x, 211xl2y, 2llyl2y (multiquantum coherence, not observable27 spins. All observable (and non-observable) physical values can be extracted by multiplying the density matrix with their appropriate operator and then calculating the trace of the resulting matrix. The time-dependent evolution of the system is calculated by unitary transformations (corresponding to "rotations") of the density matrix operator with the proper Hamiltonian H (including r.f. pulses, chemical shift evolution, J coupling etc.): r(t') = exp{iHt} r(t) exp{-iHt} (for calculations these exponential operators have to be expanded into a Taylor series). The density operator can be written als linear combination of a set of basis operators. Two specific bases turn out to be useful for NMR experiments: - the real Cartesian product operators Ix, Iy and Iz (useful for description of observable magnetization and effects of r.f. pulses, J coupling and chemical shift) and - the complex single-element basis set I+ , I- , Ia and Ib (raising / lowering operators, useful for coherence order selection / phase cycling / gradient selection). Cartesian Product operators Lit. O.W. Sørensen et al. (1983), Prog. NMR. Spectr. 16, 163-192 Single spin operators correspond to magnetization of single spins, analogous to the classical macroscopic magnetization Mx , My , Mz . Ix , Iy (in-phase coherence, observable) Iz (z polarisation, not observable) Two-spin operators 2I1xI2z , 2I1yI2z , 2I1zI2x , 2I1zI2y (antiphase coherence, not observable) 2I1z I2z (longitudinal two-spin order, not observable) 2I1xI2x , 2I1yI2x , 2I1xI2y , 2I1yI2y (multiquantum coherence, not observable)