Читать книгу Essential Concepts in MRI - Yang Xia - Страница 38

In Appendix A2.6, we introduce the concept of density matrix operator ρ, which relates the expectation value of any operator A to the trace of the matrix product Aρ, via Eq. (A2.33). Since Ix=12[0110] in the formalism of Pauli’s spin matrices (cf. Appendix A2.5), we have, from Eq. (3.12),

(3.16)

where Tr represents the trace of the matrix (the sum of the diagonal elements) – see Eqs. (A2.32) and (A2.33), and the two examples at the end of Appendix A2.6. The term in brackets describes the degree of “single quantum coherence” of the ensemble, while the average (the bars) reflects the phase coherence between the +1/2 and –1/2 states. At thermal equilibrium, both a1/2*a−1/2¯ and a1/2a−1/2*¯ are zero (no transverse component), while the term (|a1/2|2¯−|a−1/2|2¯) has a stable value (the longitudinal magnetization along the z axis).