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, I y, I z 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 (I x, I y, and I z ): The density operator of a spin-1/2 system:  (t) = a(t)T x + b(t)I y + c(t)I z At equilibrium only I z is non-vanishing. Thus,  (t) eq = I z with c(t) = 1.

Hamiltonian for free precession (During delay time): H =  ·I z, where  is the rotational frequency. Hamiltonian of an X-pulse: H =  1 ·I x. Similarly, H =  1 ·I y 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 =  1 I x ;  (0) = I z., Thus,  (t) = exp(-iHt)  (0)exp(iHt) = exp(-i  1t p I x )I z exp(i  1t p I x ) = I z Cos  1 t p – I y Sin  1 t p Standard rotations: exp(-i  I a )(old operator)exp(i  I a ) = cos  (Old operator) + sin  (new operator) Example: exp(-i  I x )I y exp(i  I x ) Old operator = I y and new operator = I z  Find new operator  xp(-i  I x )I y exp(i  I x ) = cos  I y + sin  I z

Example 2: exp(-i  I y ){-I z }exp(i  I y ) = -cos  I z - sin  I x Shorthand notation:  (t p ) = exp(-i  1 t p I x )  (0) exp(i  1 t p I x )  For the case where  (0) = I z, Example: Calculate the results of spin echo At time 0:  (0) = I z; At time 0  a: Rotation about x by 90 o :  (0) = - I y At time a  b: Free precession (Operator =  I Z ) (Cosidered as rotation wrt Z-axis) At b  c rotation wrt x by 180 o : The second term is not affected Or Thus:   90 x 180 x 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 I x and I y we got (cos  cos  + sin  sin  )T y + (cos  sin  - sin  cos  ) I x = I y Or  Independent of  and   The magnetization is refocused.  -   - refocus the magnetization and is equivalent to pulse. Product operator for two spins: Cannot be treated by vector model Two pure spin states: I 1x, I 1y, I 1z and I 2x, I 2y, I 2z I 1x and I 2x are two absorption mode signals and I 1y and I 2y are two dispersion mode signals. These are all observables (Classical vectors)

Coupled two spins: Each spin splits into two spins Anti-phase magnetization: 2I 1x I 2z, 2I 1y I 2z, 2I 1z I 2x, 2I 1z I 2y (Single quantum coherence) (Not observable) Double quantum coherence: 2I 1x I 2x, 2I 1x I 2y, 2I 1y I 2x, 2I 1y I 2y (Not directly observable) Zero quantum coherence: I 1z I 2z (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 I 1x of a coupled two-spin system: Hamiltonian: H free =  1 I 1z +  2 I 2z = cos  1 tI 1x + sin  1 tI 1y No effect Example 2: The evolution of 2I 1x I 2z under a 90 o pulse about the y-axis applied to both spins: Hamiltonian: H free =  1 I 1y +  1 I 2y

Evolution under coupling: Hamiltonian: H J = 2  J 12 I 1z I 2z 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 I x 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  = 1/4J I x completely converts to antiphase 2I y I z.  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 I x 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(I x + iI y ); I - = 1/2 (I x –i - I y ) Coherence order of I + is p = +1 and that of I - is p = -1  I x = ½(I + + I - ); I y = 1/2i (I + - I - ) are both mixed states contain order p = +1 and p = -1 For the operator: 2I 1x I 2x we have: 2I 1x I 2x = 2x 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+ ) The double quantum part, ½(I 1+ I 2+ + I 1- I 2- ) can be rewritten as: Similar the zero quantum part can be rewritten as: ½(I 1+ I 2- + I 1- I 2+ ) = ½ (2I 1x I 2x – 2I 1y I 2y ) P = +2 P = -2P = 0 (Pure double quantum state) (Pure zero quantum state)

Three spin system: Total number of base operators = Unitary OP + 3x3 one- spin OP + 3x3x3 two-spin OP three spin OP = 64 Consider the case of spin 1 coupled to spins 2 and 3 with coupling constants J 12 and J 13 where J 12 > J 13. the rule governing the evolution of spins under chemical shift evolution, pulse effect and coupling effect are the same as in the case of two spin system. Operator of spin 1 affect only spin 1, but not spins 2 or 3. Similarly we consider effect of J 12 coupling only on spins 1 and 2, but not on 3 and vise versa. Examples:

Multiple Quantum Coherence: Active spins: Spins that contains transverse components, I x or I y. Passive spins: Spins that contain only the longitudinal component, I z. Evolution of Multiple Quantum Coherence: Chemical shift evolution: Analogous to that of I x and I y except that it evolves with frequency of  1 +  2 for p = ±2 and  1 -  2 for p = 0

Coupling: