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The pulse is applied mathematically by multiplying the spin state matrix a by the rotation matrix R and then multiplying this result by the inverse matrix R-1 (the product R-1 R is the identity matrix 1). For rotation (pulse) operators, the inverse matrix is simply the rotation in the opposite direction (©' = -©). Note that the final result is the same as the representation of the product operator Ix given above.

Matrix multiplication involves forming a sum of products of matrix elements. For example, in the final multiplication above, the upper right-hand element of the product matrix

(C) is formed from the sum (1 x 1 + (—1)x(-1)) = 2. The first 1 and —1 came from the first row of the left-hand matrix (A) and the second 1 and — 1 came from the second column of the second matrix (B). The 2 was then factored out of the entire product matrix to change the 1/4 to a 1/2. In general, the element in the ith row and the jth column of the product matrix is formed from the Ith row of the first matrix and the jth column of the second matrix by forming a sum of products of elements: anbij + ai2b2j +----+ ainbnj, where an is the first element in the ith row of the first matrix and bij is the first element in the jth column of the second matrix. Matrix multiplication does not generally give the same result if the order of the two matrices being multiplied is reversed; for this reason the order is important. The identity matrix (1) is a square n x n matrix with a 1 for each diagonal element 1ii and a zero for each off-diagonal element 1ij (i = j).

What is the effect of a time delay on the density matrix? Each off-diagonal element gets multiplied by a phase factor that depends on the length of time of the delay and the energy difference between the two energy levels that are connected by that transition:

phase factor = eimt, where m = 2nvo The exponent of an imaginary number is a shorthand for two trigonometric functions:

Note that m is just the Larmor frequency and, because real numbers are associated with the x axis and imaginary numbers with the y axis, time evolution is simply rotation in the x-y plane at the offset frequency. For double-quantum transitions, m = mI + mS, and for zero-quantum transitions m = mI — mS. For example, a 90° pulse on the y' axis followed by a delay t would give

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