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Chapter 6 Product Operator Product operator is a complete and rigorous quantum mechanical description of NMR experiments and is most suited in describing multiple pulse experiments. Wave functions: Describe the state of a system. One can calculate all

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## Chapter 6 Product Operator

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**Chapter 6 Product Operator**Product operator is a complete and rigorous quantum mechanical description of NMR experiments and is most suited in describing multiple pulse experiments. Wave functions: Describe the state of a system. One can calculate all properties of the system from its wavefunction. Operators: Represents an observable which operate on a function to give a new function. Angular Momentum: A measure of the ability of an object to continue rotational motion. Spin angular momentum Ix, Iy, Iz etc. Hamiltonian: The operator of energy if a system. One Hamiltonian to describe a particular interaction. The evolution of a Hamiltonian determines the state of the spins and the signal we detect. The operator of a single spin (Ix, Iy, and Iz): The density operator of a spin-1/2 system: (t) = a(t)Tx + b(t)Iy + c(t)Iz At equilibrium only Iz is non-vanishing. Thus, (t)eq = Iz with c(t) = 1 .**Hamiltonian for free precession (During delay time):**H = ·Iz, where is the rotational frequency. Hamiltonian of an X-pulse: H = 1·Ix. Similarly, H = 1·Iy is the Hamiltonian of a Y-pulse. Equation of motion of a density operator: (t) = exp(-iHt) (0)exp(iHt) Example: The effect of X-pulse to the spin in equilibrium. (Considered to be very short so that evolution caan be ignored): H = 1Ix; (0) = Iz. , Thus, (t) = exp(-iHt) (0)exp(iHt) =exp(-i 1tpIx)Izexp(i 1tpIx) = Iz Cos1tp – Iy Sin1tp Standard rotations: exp(-iIa)(old operator)exp(iIa) = cos (Old operator) + sin (new operator) Example: exp(-iIx)Iyexp(iIx) Old operator = Iy and new operator = Iz Find new operator xp(-iIx)Iyexp(iIx) = cos Iy + sin Iz**Example 2: exp(-iIy){-Iz}exp(iIy) = -cos Iz -**sin Ix Shorthand notation: (tp) = exp(-i1tpIx) (0) exp(i1tpIx) For the case where (0) = Iz, Example: Calculate the results of spin echo At time 0: (0) = Iz; At time 0 a: Rotation about x by 90o: (0) = - Iy At time a b: Free precession (Operator = IZ) (Cosidered as rotation wrt Z-axis) At b c rotation wrt x by 180o : The second term is not affected Or Thus: 90x 180x Time: 0 a b c d**C d: Free precession. Again, consider the two terms**separately we got: • First term: • Second s=term: • Collecting together the terms in Ix and Iy we got • (coscos + sinsin)Ty + (cossin - sin cos ) Ix = Iy • Or • Independent of and The magnetization is refocused. • - - 1800 - - refocus the magnetization and is equivalent to -1800 pulse. Product operator for two spins: Cannot be treated by vector model Two pure spin states: I1x, I1y, I1z and I2x, I2y, I2z I1x and I2x are two absorption mode signals and I1y and I2y are two dispersion mode signals. These are all observables (Classical vectors)**Coupled two spins:Each spin splits into two spins**Anti-phase magnetization: 2I1xI2z, 2I1yI2z, 2I1zI2x, 2I1zI2y (Single quantum coherence) (Not observable) Double quantum coherence: 2I1xI2x, 2I1xI2y, 2I1yI2x, 2I1yI2y (Not directly observable) Zero quantum coherence: I1zI2z (Not directly observable) Including an unit vector, E there are a total of 16 product operators in a weakly-coupled two-spin system. Understand the operation of these 16 operators is essential to understand multiple NMR expts.**Example 1: Free precession of spin I1x of a coupled two-spin**system: Hamiltonian: Hfree = 1I1z + 2I2z = cos1tI1x + sin1tI1y No effect Example 2: The evolution of 2I1xI2z under a 90o pulse about the y-axis applied to both spins: Hamiltonian: Hfree = 1I1y + 1I2y**Evolution under coupling:**Hamiltonian: HJ= 2J12I1zI2z Causes inter-conversion of in-phase and anti-phase magnetization according to the Diagram, i.e. in anti and anti according to the rules: Must have one component in the X-Y plane !!!**Useful identify:**• Spin echo in homonuclear-coupled two spins: • Non-selective pulse: • Assuming only Ix present at the beginning: Since chemical shift is refocused in spin-echo expt we consider only effect of coupling and 180o pulse: • Coupling: • 180o pulse: • No effect on the magnetization if both spins are flipped by 180o !!! • The final results • When = 1/4J Ix completely converts to antiphase 2IyIz. • Used in HSQC experiment.**Inter-conversion of in-phase and anti-phase magnetizations:**In Anti: Anti in: Heteronuclear coupling: In this case one can apply the pulse to either spins such as in the sequence a c. Sequence a is similar to that of homonuclear coupling. In sequence b the 180o pulse apply only to spin 1: During second delay the coupling effect gives: Collecting terms results in only Ix left J-coupling has been refocused (So is sequence c) (No transfer of magnetization or decoupling) 180X**Coherence order:**• Raising and lowering operators: I+ = ½(Ix + iIy); I- = 1/2 (Ix –i-Iy) • Coherence order of I+ is p = +1 and that of I- is p = -1 • Ix = ½(I+ + I-); Iy = 1/2i (I+ - I-)are both mixed states contain order p = +1 and p = -1 For the operator: 2I1xI2x we have: 2I1xI2x = 2x ½(I1+ + I1-) x ½(I2+ + I2-) = ½(I1+I2+ + I1-I2-) + ½(I1+I2- + I1-I2+) The double quantum part, ½(I1+I2+ + I1-I2-) can be rewritten as: Similar the zero quantum part can be rewritten as: ½(I1+I2- + I1-I2+) = ½ (2I1xI2x – 2I1yI2y) P = +2 P = -2 P = 0 P = 0 (Pure double quantum state) (Pure zero quantum state)**Multiple Quantum Coherence:**Active spins: Spins that contains transverse components, Ix or Iy. Passive spins: Spins that contain only the longitudinal component, Iz. Evolution of Multiple Quantum Coherence: Chemical shift evolution: Analogous to that of Ix and Iy except that it evolves with frequency of 1 + 2 for p = ±2 and 1 - 2 for p = 0

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