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Product_Operators_10

# Product_Operators_10 - Density Matrix in Product Operator...

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Density Matrix in Product Operator Form BCMB/CHEM 8190

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Product Operators: Connection to Density Matrix Properties For small deviations it is convenient to work with a deviation density matrix, | σ | = | ρ | - 1/Z| E | (identity mat) • | σ | is also just a collection of numbers; product operators are specific subsets of these numbers. Some subsets are associated with observables; ie. σ 12 , σ 21 , σ 34 , σ 43 , dictate Mx and My. Some sets transform cleanly to other sets under rf pulses; ie. σ 11 , σ 22 -> σ 12 , σ 21 under a 90°x pulse (actually linear combinations of these)
Expressing σ in a basis set of matrices Rationale: collecting elements with common evolution and transformation properties σ (t) = Σ s b s (t) B S One option, the Cartesian set of spin operators B S = 2 (q-1) π k=1 N ( I kv ) a(sk) q = 0 …N; (# spin operators in product) v = x,y,z; k = nucleus index; a(sk) = 1 for q nuclei, 0 for others ( I 0 is identity, E )

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Example: a single spin ½ nucleus Four element basis set: q = 0; ½ E q = 1; I 1X , I 1Y , I 1Z Can represent any density matrix as a linear combination of these (Hemitian; σ ij = σ ij * ; σ ii real) ½ E = I Z = I Y = I X =
Representation of some simple density matrices What about σ eq ? What about σ after a π /2 pulse on +X axis? Note the simple conversion of one product operator (part of a density matrix) to another operator (part of a density matrix) under an rf pulse

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Product_Operators_10 - Density Matrix in Product Operator...

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