Chapter 6 Product Operator Product operator is a

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 I x, 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(-i. Ht) (0)exp(i. Ht) Example: The effect of X-pulse to the spin in equilibrium. (Considered to be very short so that evolution caan be ignored): H = 1 Ix; (0) = Iz. , Thus, (t) = exp(-i. Ht) (0)exp(i. Ht) = exp(-i 1 tp. Ix)Izexp(i 1 tp. Ix) = Iz Cos 1 tp – Iy Sin 1 tp Standard rotations: exp(-i Ia)(old operator)exp(i Ia) = cos (Old operator) + sin (new operator) Example: exp(-i Ix)Iyexp(i Ix) Old operator = Iy and new operator = Iz Find new operator xp(-i Ix)Iyexp(i Ix) = cos Iy + sin Iz

Example 2: exp(-i Iy){-Iz}exp(i Iy) = -cos Iz - sin Ix Shorthand notation: (tp) = exp(-i 1 tp. Ix) (0) exp(i 1 tp. Ix) For the case where (0) = Iz, Example: Calculate the results of spin echo 90 x 180 x Time: 0 a At time 0: (0) = Iz; b c d At time 0 a: Rotation about x by 90 o : (0) = - Iy At time a b: Free precession (Operator = IZ) (Cosidered as rotation wrt Z-axis) At b c rotation wrt x by 180 o : The second term is not affected Or Thus:

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 (cos + sin )Ty + (cos sin - 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: I 1 x, I 1 y, I 1 z and I 2 x, I 2 y, I 2 z I 1 x and I 2 x are two absorption mode signals and I 1 y and I 2 y are two dispersion mode signals. These are all observables (Classical vectors)

Coupled two spins: Each spin splits into two spins Anti-phase magnetization: 2 I 1 x. I 2 z, 2 I 1 y. I 2 z, 2 I 1 z. I 2 x, 2 I 1 z. I 2 y (Single quantum coherence) (Not observable) Double quantum coherence: 2 I 1 x. I 2 x, 2 I 1 x. I 2 y, 2 I 1 y. I 2 x, 2 I 1 y. I 2 y (Not directly observable) Zero quantum coherence: I 1 z. I 2 z (Not directly observable) Including an unit vector, E there 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 I 1 x of a coupled two-spin system: Hamiltonian: Hfree = 1 I 1 z + 2 I 2 z No effect = cos 1 t. I 1 x + sin 1 t. I 1 y Example 2: The evolution of 2 I 1 x. I 2 z under a 90 o pulse about the y-axis applied to both spins: Hamiltonian: Hfree = 1 I 1 y + 1 I 2 y

Evolution under coupling: Hamiltonian: HJ = 2 J 12 I 1 z. I 2 z 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 180 o pulse: Coupling: 180 o pulse: No effect on the magnetization if both spins are flipped by 180 o !!! The final results When = 1/4 J Ix completely converts to antiphase 2 Iy. Iz. 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 180 o 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) 180 X

Coherence order: Raising and lowering operators: I+ = ½(Ix + i. Iy); 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 = 1/2 i (I+ - I-) are both mixed states contain order p = +1 and p = -1 For the operator: 2 I 1 x. I 2 x we have: 2 I 1 x. I 2 x = 2 x ½(I 1+ + I 1 -) x ½(I 2+ + I 2 -) = ½(I 1+I 2+ + I 1 -I 2 -) + ½(I 1+I 2 - + I 1 -I 2+) P = +2 P = -2 P=0 The double quantum part, ½(I 1+I 2+ + I 1 -I 2 -) can be rewritten as: (Pure double quantum state) Similar the zero quantum part can be rewritten as: ½(I 1+I 2 - + I 1 -I 2+) = ½ (2 I 1 x. I 2 x – 2 I 1 y. I 2 y) (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

Coupling: