solution to [2.53] by simple differentiation:
dðtÞ
dt
¼iHexpðiHtÞð 0 ÞexpðiHtÞþexpðiHtÞð 0 ÞiHexpðiHtÞ
¼ifexpðiHtÞð 0 ÞHexpðiHtÞHexpðiHtÞð 0 ÞexpðiHtÞg
¼ifðtÞHHðtÞg¼i½H,ðtÞ:
½ 2 : 56
For completeness, some additional properties of the exponential
operator are given here. First, in the eigenbase of A, the matrix
representation of the exponential operator is
hjmexpðÞAjin ¼hjmEjin þhjmAjin þðÞ 1 = 2 hjmAAjin þ...
¼m,n 1 þAmmþðÞ 1 = 2 A^2 mmþ... ¼m,nexpðAmmÞ¼m,nexpðÞm, ½ 2 : 57 in whichm¼Amm are the eigenvalues of A. Thus, the exponential
matrix is diagonal in the eigenbase ofAand the diagonal elements
are the exponentials of the eigenvalues of A. Second, the Baker–
Campbell–Hausdorff (BCH) relationship states that
expfgA expfg¼B expAþBþ 21 ½þB,A 121 ðÞþ½þB,½B,A ½½B,A,A ...
:
½ 2 : 58 An extremely important corollary to this theorem states that
exp(AþB)¼exp(A) exp(B) if and only if [A,B]¼0( 5 ).
2.2.4 THEROTATINGFRAMETRANSFORMATION
The solution to the Liouville–von Neumann equation is straight-
forward if the Hamiltonian is time independent. A pulse sequence gen-
erally consists of two distinct parts: pulses (during which one or more
rf fields are applied) and delays (during which no rf fields are present).
For the present treatment, the time-dependent effects of the coupling
between the spin system and the lattice will be neglected; these effects
give rise to spin relaxation phenomena that will be discussed in
Chapter 5. With this simplification, the Hamiltonian governing the
delays is time independent; however, the rf fields comprising the pulses
remain time-dependent perturbations. The simplest solution to this
complication is to find a transformation that renders the rf Hamiltonian
time independent and then apply [2.54]. The transformation that renders
2.2 THEDENSITYMATRIX 43