Density_Matrix_10

Density_Matrix_10 - Density Matrix Description of NMR...

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Density Matrix Description of NMR BCMB/CHEM 8190
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Some Sequences Need Quantum Explanations For example: the INEPT and HSQC Experiments Solution: density matrix and product operator descriptions 15 N 1 H 1 H 15 N 90x 180y 90y 90x x y z x y z x y z x y z x y z x y z J-> J->
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Operators in Matrix Notation If we stay with one basis set, properties vary only because of changes in the coefficients weighting each basis set function μ x = γ (h/2 π )< ψ | Ix | ψ > ψ = c1 αα + c2 αβ + c3 βα + c4 ββ = j c j φ j < ψ | Ix | ψ > = j,k c j * c k < φ j | Ix | φ k > We need calculate < φ j | Ix | φ k > only once if we stay with this basis set – these can be put in a n x n matrix. Matrix equivalent: < ψ | Ix | ψ > = (c1, c2, …)* Ρ Τ Λ c1 Ξ Ο Ix Ο Ο c2 Ο Σ Φ Ν Π
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Special Case: Pauli Spin Matrices 0 ½ ½ 0 |Ix| = Note: < α | Ix | α > = ½ < α | β > = 0 < α | Ix | β > = ½ < α | α > = ½ 0 -i ½ i ½ 0 |Iy| = ½ 0 0 ½ |Iz| = How do they work? Try something we know: Ix | α > = ½ β 0 ½ ½ 0 1 0 0 ½ 0 1 = = ½ = ½ β Operators are a matrix of numbers, Spin functions a vector of numbers
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Density Matrices and Observables for an Ensemble of Spins Expectation values pertain to single spin properties; we observe net behavior of an ensemble of spins. For a particular system (two spin ½ nuclei) operator representations are the same; all variations are in basis set coefficients. We need to average over these coefficients. < ψ | Ix | ψ > = (c1, c2, …)* Ρ Τ Λ c1 Ξ Ο Ix Ο Ο c2 Ο Σ Φ Ν Π = j,k c j * c k Ix jk Products of averaged coefficients can also be collected and used in matrix form.
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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 Tech.

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Density_Matrix_10 - Density Matrix Description of NMR...

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