Product_Operators_10

# Product_Operators_10 - Density Matrix in Product Operator...

This preview shows pages 1–6. Sign up to view the full content.

Density Matrix in Product Operator Form BCMB/CHEM 8190

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 =
density matrices • What about σ eq ? • What about σ after a π /2 pulse on +X axis? • Note the simple conversion of one product

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 16

Product_Operators_10 - Density Matrix in Product Operator...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online