COSY Two RF pulses 90 t 1 dt

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COSY , Two RF- pulses 90° t 1 dt increments acquisition dt dwell time

COSY , Two RF- pulses 90° t 1 dt increments acquisition dt dwell time 90° t 1 –evolution, t 1 incremented cos w t 1 sin w t 1 t 2 – evolution, acquisition M+ = Mx+ i My sin w t 1 (exp (i w t 2 )) Ix- magnetization

Two Frequency Axes preparation l = t 1 = n Dt t 1 rf

Two Frequency Axes preparation l = t 1 = n Dt t 1 rf acquisition t 1 has to be incremented t 2 w w rf FT along t 1 FT along t 2 w f 1 w (Sin (w t 1 ) Sin (J t 1 )) t 1 t 2 sin ( w t 1 ) exp ( i w t 2 ) f 2

COSY – Experiment (1 H, 1 H) COSY- Experiment 90° 1 1 90° (

COSY – Experiment (1 H, 1 H) COSY- Experiment 90° 1 1 90° ( H, H) t 1 incremented t 2 acquisition propionic acid

J Coupling Evolution B 0 evolution My Ix Iy Iy w - w 0

J Coupling Evolution B 0 evolution My Ix Iy Iy w - w 0 large Ix Iy Mx Ix My Iy w - w 0 0 M x Ix Ö(Ix 2 + Iy 2)

J Evolution 2 Spins ½ Ix Iy Kopplungsentwicklung 2 Spins ½ X Nuc (31

J Evolution 2 Spins ½ Ix Iy Kopplungsentwicklung 2 Spins ½ X Nuc (31 P e. g. )

Different Viewpoints Nobel price 1952 Felix Bloch 1905 - 1983 Classical macroscopic magnetization vectors

Different Viewpoints Nobel price 1952 Felix Bloch 1905 - 1983 Classical macroscopic magnetization vectors Faraday induction Edward Mills Purcell 1912 - 1997 Quantum mechanical Stimulated (or spontaneous) absorption and Emission

Different Viewpoints D. I. Hoult and N. S. Ginsberg By the middle of the

Different Viewpoints D. I. Hoult and N. S. Ginsberg By the middle of the 20 th century, the quantum Nobel price 1952 theories of matter and its interaction with radiation were well established (1). Thus with the discovery of NMR by Bloch and Purcell, a quantum explanation of the voltage present in a receiving coil was sought. However, such an explanation was not readily forthcoming. Bloch had described his observations in terms of Faraday induction while Purcell saw his as absorption and emission of energy, and two initially had great difficulty believing that they were observing the same phenomenon. In particular, describing the origins of the free induction decay. Mills Purcell Edward Felix Bloch caused significant problems, and as the proffered explanation 1912 - 1997 1905 - 1983 of this phenomenon is still only well known to a handful of NMR physicists, we therefore begin by describing its origins Bloch und Purcell Classical macroscopic magnetization vectors Faraday induction Quantum mechanical Stimulated (or spontaneous) absorption and Emission

Different Viewpoints Klassisch vs. quantenmechanisch

Different Viewpoints Klassisch vs. quantenmechanisch

QM for Ensembles a, b F = a+b Single particle state : Wave functions

QM for Ensembles a, b F = a+b Single particle state : Wave functions as linear combination of Determine initial states : Probability(a) Fa = a , Probability(b ) F b = b d/dt F = State evolution during exp. : ih HF solve for H 1 during t 1 , H 2 during t 2 …. . Measure outcome for initial states : M x (Fa ) = < Fa | Ix | Fa > Calculate ensemble average M x = Probability(a) M x (Fa ) + Probability(b) M x (F b ) : , M x (F b ) = < F b | Ix | Fb > Different Approach : F Identify state with operator projector a OF if OF F = o. F F (1/2 + I z ) because ( 1/2 + I z ) a = a Arbitrary state linear combination : Determine initial state : State evolution during exp. : Decompose state into basis : b , OF Y = 0 < F Y > = 0 Basis : I z , Iy , Ix , 1 (1 /2 – I z ) because ( 1/2 - I z ) b = b s = c z I z , + c y I y + c x I x + c 1 1 s = Probability(a) ( ½ + I z ) + Probability(b ) for ( ½ - I z ) = I z d/dt s = i h [ H , s] s (t) = cz(t) Iz , + cy(t) Measure outcome for initial state : Mx cx(t) solve for H 1 during t 1 , H 2 during t 2 …. . Iy + cx(t) Ix + c 1(t) 1

Solution for d/dt s = i h [ H , s ] Classical mechanics

Solution for d/dt s = i h [ H , s ] Classical mechanics : Solution to dl/dt = torque by diagonalization : calculate M+ and M- rotations (precession) about axis of torque Mx (t) a Mx (0) cos ( w t ) + My (0) sin ( w t ) M+ (t) a M+ (0) exp( - i w t) QM wave function : Solution for d/dt F = ih. HF Statistical operator : Solution of d/dt s = i h [ H , s] s (t) = exp (i h H t ) s(0) s(t) = exp(-i h. H t) s(0) exp( i h. H t) Meaning of exp (operator) : exp ( c H t) = 1 + 1/1! H t c + 1/2! H 2 t 2 c 2 + 1/3! H 3 t 3 c 3 + … d/dt exp(c H t) = c H exp( c H t) = exp( c H t) c H H : constant d/dt (exp(-i H t) s(0) exp( i H t)) = -i H exp(-i H t) s(0) exp( i H t) + exp(-i H t) s(0) i H exp( i H t) d/dt (exp(-i H t) s(0) exp( i H t)) = -i H exp(-i H t) s(0) exp( i H t) + exp(-i H t) s(0) exp( i H t) ) i H d/dt (exp(-i H t) s(0) exp( i H t)) = -i H s(t) + s(t) i H d/dt (exp(-i H t) s(0) exp( i H t)) = i [ H , s(t) ] = d/dt s(t)

Auxiliary Calculation : exp(c A) exponential function exp ( c A ) = 1

Auxiliary Calculation : exp(c A) exponential function exp ( c A ) = 1 + 1/1! A c + 1/2! A 2 c 2 + 1/3! A 3 c 3 + … c =i f f real A 2 multiple of unity exp ( i f A ) = 1 + 1/1! i f A - 1/2! f 2 a 2 1 - A 2 = a 2 * 1 n! factorial of n 1/3! i f 3 a 2 A + 1/4! f 4 a 4 1 + 1/5! i f 5 a 4 A real part Re ( exp ( i f A ) ) = 1 1 - 1/2! (f a ) 2 1 + 1/4! ( f a ) 4 1 - 1/6! (f a ) 6 1 … cos ( a f ) + 1/4! ( f a )4 - 1/6! . . 1/3! (f a ) 3 i /a A + 1/5! (f a) 5 i /a A = 1 Re ( exp ( i f A ) ) = - 1/2! (f a )2 (f a )6 cos (f a ) 1 imaginary part Im ( exp ( i f A ) = 1/1! (f a ) i/a A - sin ( a f ) - 1/3! (f a )3 = 1/1! (f a ) Im ( exp ( i f A ) ) = sin (f a ) i /a A exp( i f A ) = cos ( f a ) 1 + i /a sin ( f a ) A exp( i f A ) * = cos ( f a ) 1 - i /a sin ( f a ) A + 1/5! ( f a ) 5 …. . …

Statistical Operator Initial state : s(0) = exp ( - H /k. T )

Statistical Operator Initial state : s(0) = exp ( - H /k. T ) = 1 - H /k. T + ½ (H /k. T)² : s(0) = 1 - h (1+ d) B 0 /k. T Iz …. = evolution during r. f. : w 1 tpulse = f H = B 1 Iy …. . Iz Iy ² = ¼ 1 s(f) = (cos ( f /2 ) 1 + 2 i sin ( f /2) Iy ) Iz (cos ( f /2 ) 1 - 2 i sin ( f /2 ) Iy ) s(f) = (cos ( f /2 )² - sin ( f /2 )² ) Iz + 2 (cos ( f /2 )sin ( f /2)) s(f) = cos ( f ) Iz + sin ( f ) Ix Iz cos (f ) + Ix sin (f ) Iz a Free precession Ix : H = - h (1+ d) B 0 Iz exp( i w Iz t) = cos ( w/2 t ) 1 H = w Iz + H 2 = w 2 Iz 2 = w 2 ¼ 1 i 2 sin (w/2 t ) Iz s(t) = (cos ( w/2 t ) 1 + 2 i sin(w/2 t) Iz) Ix (cos( w/2 t ) 1 - 2 i sin (w/2 t) Iz ) Ix a Ix cos (w t ) + Iy sin (w t) Measure outcome for initial states : M x Ix M x = cos (w t ) My Iy M y = sin (w t ) M +/- I+/- M +/- = exp (+/- i w t )

Two Spin System Single particle states : F linear combination of basis aa ,

Two Spin System Single particle states : F linear combination of basis aa , ab , ba, bb Initial state s e. g. F = b (a + b) = Iz + Sz If there is no coupling spins evolve independently [ A, B ] = 0 e. g. a exp( i (A + B) t) = exp( i A t) exp( i B t) = exp( i B t) exp( i A t) r. f. pulse H = w 1 I y + w 1 S y = w 1 ( I y + S y ) , s = exp( i Iy f ) exp( i Sy f ) (Iz + Sz ) exp( -i Iy f ) exp( -i Sy f ) s = exp( i Iy f ) Iz exp( -i Iy f ) + exp( i Sy f ) Sz exp( -i Sy f ) s = [ Iy , Sy ] = 0 Ix + Sx Scalar Coupling (Quantum Mechanical) Hamiltonian for time evolution AX –approximation H = w. I I z + w. S S z + J ( I x S x + I y S y + I z S z ) H = w. I I z + w. S S z + J I z S z H 2 = J 2 Iz 2 Sz 2 = J 2 1/16 1 1 exp( i J Iz Sz t) = cos ( J/4 t ) 1 + 4 i sin (J/4 t ) Iz. Sz Ix a (cos ( J/4 t ) 1 + 4 i sin (J/4 t) Iz. Sz) Ix (cos ( J/4 t ) 1 - 4 i sin (J/4 t) Iz. Sz ) Ix a Ix cos (J/2 t ) + Iy Sz sin ( J/2 t ) ? Iz Sz Ix = i Iz Ix Sz = Iy. Sz , Ix Iz Sz = -i Iy. Sz

Two Spin System Single particle states : F linear combination of basis aa ,

Two Spin System Single particle states : F linear combination of basis aa , ab , ba, bb Ix Iy Iz Ix. Sz Iy. Sz Iz. Sz Ix. Sx Sx Sy Sz Sx. Iz Sy. Iz Iy. Sx Ix. Sy e. g. F = b (a + b) Iy. Sy 1 only Ix , Sx , Iy , Sy , Iz , Sz can be observed [ A, B ] = 0 a exp( i (A + B) t) = exp( i A t) exp( i B t) = exp( i B t) exp( i A t) [ Iz , Iz Sz ] = 0 a exp( i H t) = exp( i w. I Iz t) exp( i w. S Sz t) exp( i J Iz Sz t) Time evolution according to B 0 for spins I and S and Iz. Sz can be calculated one by one. [ H, A ] = 0, [ H, C ] = 0 a exp( i H t) AC exp( - i H t) = exp( i H t) A exp( - i H t) C H = w. I I z a Sz Ix cos (w. I t ) + Sz Iy sin ( w. I t ) H = w 1 I y a Sz Ix cos (f ) - Sz Iy sin ( f. I) H = w 1 S y a Sz Ix cos (f ) + Sx Ix sin ( f. I)

Different Views Macroscopic View Quantum View Mx , My , Mz components of macroscopic

Different Views Macroscopic View Quantum View Mx , My , Mz components of macroscopic magnetization Ix , Iy , Iz operators of components of macroscopic magnetization B 0 evolution Dw t Mx a Mx cos ( w t ) + My sin ( w t ) Iy a Iy cos ( w t ) - Ix sin ( w t ) My a My cos ( w t ) - Mx sin ( w t ) M+ = Mx + i My , M- = Mx - i My M- a M- exp( i w t) M+ a M+ exp( - i w t) Ix a Ix cos ( w t ) + Iy sin ( w t ) z -y I+ = Ix + i Iy -x , I- = Ix - i Iy I- a I- exp( i w t) x I+ a I+ exp( - i w t) y -z y-pulse Mz a Mz cos ( f ) + Mx sin (f) B 1 evolution x-pulse f = w 1 tpulse z Mx a Mx cos (f ) - Mz sin (f ) x-pulse Iz a Iz cos ( f ) + Ix sin (f) -y Ix a Ix cos (f ) - Iz sin (f ) x-pulse -x x Mz a Mz cos ( f ) - My sin ( f ) My a My cos ( f ) + Mz sin ( f ) y-pulse y Iz a Iz cos ( f ) - Iy sin ( f ) Iy a Iy cos ( f ) + Iz sin ( f ) -z

Time Evolution / Rotation of Operators B 0 - and B 1 - evolution

Time Evolution / Rotation of Operators B 0 - and B 1 - evolution Iz Iz Iz -Iy -Ix -Ix Ix Ix Iy Iy -Iz -Iz J coupling evolution H= Iz. Sz - 2 Ix. Sz - Ix -2 Iy. Sz - Iy Iy p. Jt 2 Ix. Sz Ix I+ a ? : Ix wt f f Iy -Iy (Ix + i Iy ) a ( Ix cos ( f) + 2 Iy. Sz sin ( f )) + i ( Iy cos ( f) - 2 Ix. Sz sin ( f )) (Ix + i Iy ) a I+ cos ( f) + 2 i I-Sz sin ( f )

B 0 - and B 1 - Evolution Iz Iz Iz -Iy B 0

B 0 - and B 1 - Evolution Iz Iz Iz -Iy B 0 - und B 1 - Entwicklung -Ix -Iy -Ix Ix f Iy -Iz Iy wt -Iz J Coupling Evolution Iz. Sz AX- Approximation - Ix -2 Iy. Sz Ix - 2 Ix. Sz - Iy 2 Iy. Sz Iy p. Jt 2 Ix. Sz Ix B 0 - and B 1 -Evolution Iz. Sz Iz -Iy. Sz -Ix. Sz Ix f Iy. Sz -Iz. Sz Iy Sz Iz ( Iy cos w. I t 1 - Ix sin w. I t 1 ) Sz Ix + S x Ix. Sz Iy. Sz ( Iy cos f + Iz sin f) S(z Sz cos f - Sy sin f ) wt -Iz

J Evolution 2 Spins ½ Ix IIxy Iy Kopplungsentwicklung 2 Spins ½ Ix Iy

J Evolution 2 Spins ½ Ix IIxy Iy Kopplungsentwicklung 2 Spins ½ Ix Iy expectation values of Iy and Ix Iy I x Sz < Iy >Ens ~ - cos ( w. I t ) cos ( J/2 t ) < Ix Sz >Ens ~ sin ( w. I t ) sin ( J/2 t ) Rotating frame of reference assumed for Spin I w. I = 0 < Iy >Ens ~ cos ( J/2 t ) , < Ix Sz >Ens ~ - sin ( J/2 t )

J Evolution B 0 (shift) evolution Iy Iy cos( w. I t ) Iy

J Evolution B 0 (shift) evolution Iy Iy cos( w. I t ) Iy + sin( w. I t ) Ix Kopplungsentwicklung Ix Ix J evolution cos( J/2 t ) Iy Iy Iy + sin( J/2 t ) Ix Sz Ix Ix

J Evolution Ö(Ix 2 + Iy 2)

J Evolution Ö(Ix 2 + Iy 2)

J Evolution B 0 z z Kopplungsentwicklung r r x x y y After

J Evolution B 0 z z Kopplungsentwicklung r r x x y y After application of a 90° y pulse only the x components of angular momentum are known. Z and y components are uncertain. QM state (no proper picture just a visualization)

J Evolution J evolution (AX- approximation): Spins undergo a precession about the z component

J Evolution J evolution (AX- approximation): Spins undergo a precession about the z component of their coupling „ partners“. Kopplungsentwicklung QM state (no proper picture just a visualization) At some time no macroscopic magnetization can be observed. J evolution carries on in the same direction QM state (no proper picture just a visualization) At some time macroscopic magnetization is restored.

J Evolution S-spin I-spin QM state (nozproper picture just a visualization) y Ix- magnetization

J Evolution S-spin I-spin QM state (nozproper picture just a visualization) y Ix- magnetization + x Sx- magnetization neither Ix- nor Iy- magnetization QM state +z -y z z (no proper picture just a visualization) -y -y z -y neither Sx- nor Sy- magnetization Iz. Sy- coherence -x +y +x z x y y x -z z x y x x Iy. Sz- coherence -Ix- magnetization + -Sx- magnetization no Iz. Sy , Iy. Sz - coherence

J Evolution / COSY Ix- magnetization Sx- magnetization Kopplungsentwicklung z z y Ein Zustand.

J Evolution / COSY Ix- magnetization Sx- magnetization Kopplungsentwicklung z z y Ein Zustand. z y Ix- magnetization z (Sx- magnetization ) y y t 1 x z z x y y x x z y z x x y x x Iy. Sz- coherence (Iy- magnetization Iy. Sz- coherence Iy- magnetization ) 90 ° Iz. Sy- coherence t 2 - acquisition Sx- magnetization Iy. Sz- coherence

COSY , Two RF- pulses 90° t 1 dt increments acquisition dt dwell time

COSY , Two RF- pulses 90° t 1 dt increments acquisition dt dwell time Sx- magnetization 90° Sy- magnetization Iz. Sy- coherence t 1 –evolution, t 1 incremented cos w t 1 Iy. Sz- coherence sin w t 1 t 2 – evolution, acquisition M+ = Mx+ i My sin w t 1 (exp (i w t 2 )) 90° Ix- magnetization

COSY Experiment

COSY Experiment

J Evolution (Ensemble Average) ensemble B 0 thermal equilibrium c. Iz = 0 after

J Evolution (Ensemble Average) ensemble B 0 thermal equilibrium c. Iz = 0 after 90° - pulse c. Ix = 0 c. Sz = 0 c. Sx = 0 after sel. -90° - pulse I nucleus c. Iz = 0 c. Sx = 0 c. Iz. Sx = 0 after J coupling evolution c. Iz = 0 c. Iz. Sx = 0 y c. Iz. Sy = 0 x c. Sx = 0

Axioms of Quantum Theory* * Frederic Schuller FAB Erlangen Nuremberg Axiom 1 : with

Axioms of Quantum Theory* * Frederic Schuller FAB Erlangen Nuremberg Axiom 1 : with every quantum system, there is associated a complex Hilbertspace (H, +, . , < , >) The states of the system are all positive trace-class linear maps r H -> H for which Tr r = 1 Almost everywhere it is stated: the (normalized) elements y e H are the states of the quantum system. False !!

Axioms of Quantum Theory* * Frederic Schuller FAB Erlangen Nuremberg Axiom 4 : Unitary

Axioms of Quantum Theory* * Frederic Schuller FAB Erlangen Nuremberg Axiom 4 : Unitary dynamics: during time intervals (t 1, t 2) during which no measurement occur state r(t 2) @ time t 2 r(t 1) @ time t 1 are related through: r(t 2)=U(t 2 -t 1) r(t 1) U-1(t 2 -t 1) where U(t) : = exp (-i / h H t ) H energy observable

Appendix 1

Appendix 1

COSY Experiment Iz. Sz t 1 acquisition t 2 Iz Iz Iz. Sz -

COSY Experiment Iz. Sz t 1 acquisition t 2 Iz Iz Iz. Sz - 2 Ix. Sz Iz - Ix -Iy -Iy. Sz thermal equilibrium - -Ix Iy Iz + Sz Iy. Sz 90° y-pulse Ix + Sx shift w. I ws p. Jt 2 Ix. Sz Ix cos w. I t 1 + Iy sin w. I t 1 Ix Iy -2 Iy. Sz -Ix 2 Iy. Sz Ix Ix p. Jt Iy Iy -Iz Ix -Iz + Sx cos w. S t 1 + Sy sin ws t 1 coupling ( Ix cos p J t 1 + 2 Iy Sz sin p J t 1 ) cos w. I t 1 + ( Iy cos p J t 1 - 2 Ix Sz sin p J t 1 ) sin w. I t 1 ( - Iz cos p J t 1 + 2 Iy Sx sin p J t 1 ) cos w. I t 1 + ( Iy cos p J t 1 + 2 Iz Sx sin p J t 1 ) sin w. I t 1 90° y-pulse only Ix + i Iy and Sx + i Sy can be recorded (observables) : Iz does not change and can‘t be recorded. how does 2 Iy Sx evolve ? ( Iy cos w. I t 2 - Ix sin w. I t 2 ) ( Sx cos w. S t 2 + Sy sin w. S t 2 ) shift w. I ws Iy Sx cos w. I t 2 cos w. S t 2 - Ix Sy sin w. I t 2 sin w. S t 2 - Ix Sx sin w. I t 2 cos w. S t 2 + Iy Sy cos w. I t 2 sin w. S t 2 )

xes ition t 1 rf t 2 t 1 h as to be i

xes ition t 1 rf t 2 t 1 h as to be i ncre men ted w rf FT along t 2 FT t 1 t 1 ) ex p(i (w 1 n t 1 ) sin ( Jt )S in (J f 1 wt 2 (S i n(w ) alon g t 1 w )) t 1 w f 2 w

COSY – Experiment (1 H, 1 H) COSY- Experiment 90° 1 1 90° (

COSY – Experiment (1 H, 1 H) COSY- Experiment 90° 1 1 90° ( H, H) t 1 incremented t 2 acquisition propionic acid

COSY Experiment Phase Cycles COSY- Experiment

COSY Experiment Phase Cycles COSY- Experiment

COSY Experiment t 1 acquisition t 2 only ( Iy cos p J t

COSY Experiment t 1 acquisition t 2 only ( Iy cos p J t 1 sin w. I t 1 + 2 Iz Sx sin p J t 1 sin w. I t 1 ) yields an observable signal. Iy cos p J t 1 sin w. I t 1 along t 1 delivers after FT in t 1 ( d(n 2 - n. I - J/2) + d(n 2 - n. I + J/2) ) cos p J t 1 sin w. I t 1 modulated by J and shift evolution according to cos p J t 1 sin w. I t 1 : delivers pure real function Iy cos p J t 1 sin w. I t 1 cos FT with respect to t 1 yields: (d(n 2 - n. I - J/2) + d(n 2 - n. I + J/2) )( d(n 1 - n. I - J/2) + d(n 1 - n. I + J/2) ) = d(n 2 - n. I - J/2) d(n 1 - n. I - J/2) + d(n 2 - n. I + J/2) d(n 1 - n. I + J/2) + d(n 2 - n. I - J/2) d(n 1 - n. I + J/2) ) + d(n 2 - n. I + J/2) ( d(n 1 - n. I - J/2) diagonal and off- diagonal peaks No plain phase a calculate absolute value (magnitude spectrum ) || spc(n 2 , n 1 ) || = Ö ( spc(n 2 , n 1 ) * ) n 1 t 1 J J n. S t 2 n. I n. S n 2 n. I n 2

COSY Experiment t 1 acquisition t 2 COSY- Experiment only ( Iy cos p

COSY Experiment t 1 acquisition t 2 COSY- Experiment only ( Iy cos p J t 1 sin w. I t 1 + 2 Iz Sx sin p J t 1 sin w. I t 1 ) ) yields an observable signal. 2 Iz Sx sin p J t 1 sin w. I t 1 a a real function of t 1 yielding after successive J- evolution Sy – magnetization. FT with respect to t 2 delivers ( d(n 2 - n. S - J/2) + d(n 2 - n. S+ J/2) ) sin p J t 1 sin w. I t 1 Sy sin p J t 2 sin p J t 1 sin w. I t 1 a with cos FT in t 1 ( d(n 2 - n. S - J/2) + d(n 2 - n. S + J/2) ) ( d(n 1 - n. I - J/2) + d(n 1 - n. I + J/2) ) = d(n 2 - n. S - J/2) d(n 1 - n. I - J/2) + d(n 2 - n. S + J/2) d(n 1 - n. I + J/2) + d(n 2 - n. S - J/2) d(n 1 - n. I + J/2) ) + d(n 2 - n. S + J/2) ( d(n 1 - n. I - J/2) cross - peaks No simple phase relation therefore a absolute value spectrum || spc(n 2 , n 1 ) || n 1 t 1 = Ö ( spc(n 2 , n 1 ) ) * J J n. S t 2 n. I n. S n 2 n. I n 2

2 D Spectrum FID ( t 1 , t 2 ) = sin w

2 D Spectrum FID ( t 1 , t 2 ) = sin w At 1 (exp (i w. A t 2 )) 2 D- Spektrum * + cos JAB/2 t 1 * sin w At 1 (exp (i w. B t 2 )) * FT : sin w. B t 1 (exp (i w. B t 2 )) + cos JAB/2 t 2 sin JAB/2 t 1 * * * cos JAB/2 t 2 sin w. B t 1 (exp (i w. A t 2 )) + sin JAB/2 t 2 SPC ( t 1 , w 2 ) = d (w. A - w 1 ) d (w. A -w 2 ) cos JAB/2 t 1 * + sin JAB/2 t 1 * sin JAB/2 t 2 d (w. B -w 1 ) d (w. B -w 2 ) * (d ( -JAB /2 ) + d ( JAB /2 )) * (d ( - JAB /2 ) + d ( JAB /2 )) w. A * (d ( -JAB /2 ) + d ( JAB /2 )) d (w. A -w 1 ) d (w. A -w 2 )

COSY Experiment t 1 acquisition t 2 COSY- Experiment How does 2 Iy Sx

COSY Experiment t 1 acquisition t 2 COSY- Experiment How does 2 Iy Sx ? evolve ? shift w. I ws Iy Sx cos w. I t 2 cos w. S t 2 - Ix Sy sin w. I t 2 sin w. S t 2 - Ix Sx sin w. I t 2 cos w. S t 2 + Iy Sy cos w. I t 2 sin w. S t 2 ) Ix Sy cos w. S t 2 cos w. I t 2 - Iy Sx sin w. S t 2 sin w. I t 2 - Ix Sx sin w. S t 2 cos w. I t 2 + Iy Sy cos w. S t 2 sin w. I t 2 ) Iy Sx ( cos w. I t 2 cos w. S t 2 - sin w. S t 2 sin w. I t 2 ) - Ix Sx ( sin w. I t 2 cos w. S t 2 + sin w. S t 2 cos w. I t 2 ) + Ix Sy ( cos w. I t 2 cos w. S t 2 - sin w. S t 2 sin w. I t 2 ) - Iy Sy (cos w. I t 2 sin w. S t 2 + cos w. S t 2 sin w. I t 2 ) cos a cos b - sin a sin b = cos ( a + b ) cos a sin b - sin a cos b = sin ( a + b ) cos a sin -b - sin a cos -b = sin ( a - b ) - cos a sin b - sin a cos b = sin ( a - b ) Iy Sx ( cos ( (w. I + w. S ) t 2 ) + Ix Sy ( cos ( (w. I + w. S ) t 2 ) double quantum coherences Ix Sx ( sin( ( w. S - w. I ) t 2 ) zero quantum coherences + Iy Sy ( sin( ( w. S - w. I ) t 2 ) J coupling evolution : Iy Sx a Iy ( Sx + S y Iz ) a ( Iy - Iy Sz ) ( Sx + S y Iz ) = … Iy Sz Sx … Iy Sz Sy Iz = … Iy Sy … Ix Sx