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)
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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 =
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density matrices • What about σ eq ? • What about σ after a π /2 pulse on +X axis? • Note the simple conversion of one product
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This note was uploaded on 11/07/2011 for the course CHEM 8853R taught by Professor Gelbaum during the Fall '11 term at Georgia Institute of Technology.

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Product_Operators_10 - Density Matrix in Product Operator...

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