QMGreensite_merged
rotating frame or laboratory frame density operator is intended. If ¼! 1 p, is defined to be the flip angle of the pulse of len ...
Tis known as the inversion operator and has the effect of changing the spin quantum number fromþ1/2 to1/2 andvice versa. This l ...
in whichnis a unit vector along the direction of the effective field in the rotating frameBr, given by [1.18], is given by [1.23 ...
In isotropic solution, the Zeeman, chemical shift, scalar coupling, and rf pulse terms are the dominant interactions. The expect ...
E/Nis a constant that does not affect the NMR experiment; accordingly, this term is normally not written explicitly and the high ...
in which the last line is obtained by using the Pauli spin matrices, [2.71]. If ¼ 1808 , the final matrix would be equal to Rxð ...
Using the Pauli spin matrices, [2.135] can be written as ðÞ¼t UðÞ 0 U^1 ¼expðÞi tIzIyexpðÞ¼i tIz IxsinðÞ t IycosðÞ t: ½ 2 ...
2.5.1 DIRECTPRODUCTSPACES The wavefunctions in the product basis are given by the direct productsof the wavefunctions for indivi ...
representation ofIzþSzin a two-spin system must be a 44 matrix because the vector space is spanned by four wavefunctions; thus, ...
a two-spin system) ABij ðÞA B jii j ¼Ajii Bj , ½ 2 : 148 in whichAis an operator that acts on theispin andBis an operator ...
in which!i is the Larmor frequency of the ith spin andJij is the scalar coupling constant between theith and jth spins. The eige ...
The wavefunctions of [2.158] are symmetric or antisymmetric under the exchange of identical particles, as is required by the pos ...
In order for 2 to be an eigenfunction, the two terms in parentheses following the last equal sign must be identical. Thus, !I! ...
Using the matrix representation given in [2.152], the following relation- ship is easily derived ð 2 IzSzÞ^2 n¼E: ½ 2 : 168 Su ...
For example, using the strong coupling eigenbasis [2.158], the matrix element½Rx^1 ðÞ 12 is calculated as Rx^1 ðÞ 12 ¼hj ...
ofRx^1 ðÞ , RxðÞ¼ c^2 icsu icsv s^2 icsu 1 s^2 u^2 s^2 uv icsu icsv s^2 uv 1 s^2 v^2 icsv s^2 icsu icsv c^2 2 6 ( ...
2.5.4 ONE-PULSEEXPERIMENT FOR ATWO-SPINSYSTEM To compute the observable magnetization following a pulse and subsequent free prec ...
because [2.143] yields the results that Iy¼ 1 2 00 i 0 00 0i i 00 0 0 i 00 2 (^66) 4 3 (^77) 5 and Sy¼ 1 2 0 i 00 i 000 000 ...
The trace of this matrix is proportional to the observed complex magnetization: Mþ ðtÞ/eið^ IþJISÞtþeið^ IJISÞtþeið^ SþJISÞt ...
T ABLE 2.2 Product Operators in the Cartesian Basis for a Two-Spin System 1 2 E ¼ 1 2 1000010000100001 2 664 3 775 Iz ¼ 1 2 10 0 ...
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