NMR for Analytical Chemists. NMR Ch Announcement. Midterm Oct 18 I will give you unknown samples for presentation next Monday

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1 Chem 56 NMR for Analytial Chemts Leture 1 NMR Ch.1-.3 Announement Midterm Ot 18 I will give you unknown samples for presentation next Monday 1

2 Quiz Q1 What the expeted motion of M(t) in the rotation frame under the following B eff? Desribe M(t) = [M x (t), M y (t), M z (t)]. eff (a) - B eff (t) = [, ω 1, ] M() = [,, M ] () Draw a trajetory of M(t) for (a). When should we stop applying a pulse to exite NMR signals? Give the time t that yields the strongest signal. What exp(-iht)? (t) = exp(-iht) () exp(iht) [.55] 3 exp(a) = 1 A A / A / 3!.. n A n / n! exp(-iγb I z t) = 1 -iγb I z t +(-iγb I z t) / + (-iγb I z t) 3 /3! + (-iγb I z t) 4 /4! +.

3 Matrix alulation of I z n I z I I I I z 3 z m z m z 1/ 1/ 1/ 1/ 1/ Q4 1 Q 1/ 1/ 1/ 4 Q 3 Q Q5 Q6 1/ 4 I z Q7 Q8 [Q] Q1 Q11 m 1/ 4 Q1 Q13 [Q3] 1 Q15 Q16 m 1/ 4 I Z Q17 Q18 [Q4] 4 exp(-iγb I z t) = 1 -iγb I z t +(-iγb I z t) / + (-iγb I z t) 3 /3! +. Using (I z ) = E exp(-iαi z ) = E +{-iα/}i z + {(-iα/) /}E + {(-iα/) 3 /3!} I z + = Eos(α/) ii z sin(α/) = [Q1 in a matrix?] exp(-iαi z ) I x exp(iαi z ) = I X osα+ I Y sinα exp(-iαi z ) I Y exp(iαi z ) = [Q1] I y osα -I X sinα 3

4 4 exp(-iαi z ) I x exp(iαi z ) = I X osα+ I Y sinα?? 1 1 / / 1/ sin os 1 i sin 1 1 os sin os i i i Expetation value in Dira notation d Ψ(t)>/dt = ih(t) Ψ(t)> In general, state ket Ψ(t)> exanded as Ψ(t)> = = Σ k (t) φ k > = Σ k (t) k >, here φ > an eigen ket (k 1 ) where φ k > an eigen ket (k =1,..) The expetation value <A> obtained by use of operator A as <A>= =< Ψ(t) A Ψ(t)> <A> given by A as ), ( * ), ( t x A t x dx <A> given by A as <A> = <Ψ(t) A Ψ(t)> = (Σ k* (t)<k )A(Σ j* (t) j>) = ΣΣ j (t) k* (t)<k A j> = ΣΣ j (t) k* (t)a kj In Dira notation, operator an be generally denoted by a matrix A kj

5 Density funtion & Expetation values id Ψ(t)>/dt = H(t) Ψ(t)> Ψ(t)> = Σ j (t) φ j > <A> = <Ψ(t) A Ψ(t)> = ΣΣ j (t) k* (t)<k A j> Density funtion approah: Density matrix for a pure state (t) Ψ(t)><Ψ(t) = Σ * k (t) j (t) j><k <A> = Tr(A(t)) = Σ<m A(t) m> m = Σ{ Σ k* (t) j (t)<m A j><k m>} k,j k,j m = Σ* k (t) j (t)<k A j> k,j Note: Tr(B) = <n B n> Ensemble and Quantum Statt If more than one systems oext, the ensemble average yields <A> = Σ w m <Ψ m (t) A Ψ m (t)> In a density funtion approah, we just need to redefine (t) for a mixed state as (t) Σ w m m = Σ w m Ψ m (t)><ψ m (t) <A> = Σw m Tr(A m (t)) =Tr(A(t)) 5

6 Density Matrix for Spin Operator Let s think about the density matrix in an equilibrium state, () ( ) for a single spin. () = P >< + P >< ~ {(1 - E /kt)/} >< + {(1 - E /kt)/} >< = E/ - ħγbb /(kt) {1/ >< - 1/ >< } = E/ - {ħγb /kt} I z Comparon: density matrix (t) and state ket (t)> Time dependene of a state ket d (t)>/dt = -ih (t)> (t)> = exp(-iht) ()> (t)> = U(t) ()> In ontrast (t) = (t)> <(t) = exp(-iht) ()><() exp(iht) = exp(-iht) () exp(iht) U = exp(-iht) alled time-evolution [Q1] operator (t) = U(t) () U (t) Time dependene of density operator governed by d(t)/dt = -i[h, (t)] (Liouville-von Neuman eq. ) 6

7 Time-Evolution of Density Matrix (t) = P m Ψ m (t)><ψ m (t) For eah Ψ m (t)>, Shrödinger equation given by d Ψ m (t)>/dt = -ih Ψ m (t)>. Th yields Ψ m (t)> = exp(-ie m t) Ψ m ()> = exp(-iht) Ψ m ()>. The orresponding ket <Ψ m (t) = <Ψ m () exp(iht) Hene, (t) =P m exp(-iht) Ψ m ()><Ψ m () exp(iht) = exp(iht){p m Ψ m ()><Ψ m () }exp(iht) = exp(-iht) () exp(iht) [.55] Motions of σ(t) d(t)/dt = -i[h, (t)] In NMR, Zeeman Interation yields E mz = -γb m z From Shrödinger Equation E > = H > (t) =exp(-iht) () exp(iht) H = -γb I z Q1. Assume σ() =I z, what happens? Q. If σ() =I X, what happens? 7

8 .7.3 Produt Operators If three operators satfy the ommutation relationship [A, B] =ic and its yli permutation exp(-ic)aexp(ic) = Aos + Bsin [I X, I Y ] = ii Z [I Z, I X ] = ii Y Rotation along C exp(-ic)aexp(ic) = Aos + Bsin 8

9 Pratie of Produt/Rotation Operators R z (φ) = exp(-iφi z ) R z (φ)i X R z (φ) = os(φ) I X + sin(φ) I Y R z (φ)i Y R z (φ) = os(φ) I Y -sin(φ) I X R X (φ) = exp(-iφi X ) R X (φ)i X R X (φ) = [Q1] R X (φ)i Y R X (φ) = os(φ) I Y + sin(φ)[q] R X (φ)i Z R X (φ) = [Q3] Rotation along C exp(-ii Z )I X exp(ii Z ) = I X os + I Y sin 9

10 Homework σ e (t) = exp{-ii X t}i Y exp {ii X t} = [Q1]ost + [Q]sin t. σ e (t) = exp{ii X t}i Y exp {-ii X t} = [Q3]ost + [Q4]sin t. σ e (t) = exp{-ii X t}i Z exp {ii X t} = [Q5]os 1 t + [Q6]sin 1 t. σ e (t) = exp{-ii Y t}i X exp {ii Y t} = [Q7]ost + [Q8]sin t. Calulation of <M z >, <M x >, <M Y > <M z > = Tr{(ħI z ) ()} = Tr{(ħI z ) E/ - ({ħγb /kt) (ħi z )I z } =. 1. Prove <M Z > = γ ħ B /4kT. (th an expeted magnetization for 1 spin). Prove <M X > = when M X = ħi X 1

11 Summary of 1D The motion of the magneti moment simply summarized as I Z [π/ I Y ] I X and I X -[t I Z ] I X os(t) + I Y sin(t). Observing NMR Signal in Quantum Mehan σ(t) = exp(-i ti z ) I x exp(i ti z ) = I X os(q1)+ I Y sin(q1) <M x (t)> = Tr{γI x σ(t)} = Tr{γI x (I X osα+ I Y sinα)} = γosαtr(i x )+ γsinαtr(i x I Y ) = [Q] <M Y (t)> = γtr{i Y σ(t)} = γtr{i Y (I X osα+ I Y sinα)} = (γosα)tr(i Y I x )+ (γsinα)tr(i Y ) = [Q3] <M + (t)> = < M x (t)> + i<m Y (t)>= [Q] + [Q3] 11

12 Home work 7A: General Formula Baker Campbell-Hausdorff lemma B = exp(-iαa) B exp(iαa) = B() + (-iα)[a, B]+ (-iα) /[A,[A, B]]+ +(-iα) 3 /3![A,[A,[A, B]]]+ [.7] (t)= exp(-iαi Z ) I X exp(iαi Z ) = I X + (-iα)[i Z, I X ]+ (-iα) /[I Z, [I Z, I X ]]+ = I X + (-iα)ii Y +(iα) (-iα) / [I Z, ii Y ]+ = I X + (-iα)ii Y + (-iα) / I X + (-iα) 3 /3! ii Y + = I X (-iα) m /(m)! + ii Y (-iα) m+1 /(m+1)! = I X [Q1] + ii Y [Q] = [Q3] Home work 7B B = exp(-iαa) B exp(iαa) B = B() + (-iα)[a, B]+ (-iα) /[A,[A, B]]+ +(-iα) 3 /3![A,[A,[A, B]]]+.. [.7] When A = I Z and B=I X C = I Y prove that [A,[A, B]] = [A,[A,[A,[A, B]]]] = [A,[A,[A,[A,[A,[A,B]]]]]]]= B and that [A,B]] =[A,[A,[A, B]]] = [A,[A,[A,[A,[A,B]]]]]= ic Prove that th orret for A = I X, B =I Y, C = I z 1

13 ..4 Rotating-frame transformation (p43) U: A unitary transformation from a laboratory frame to a new frame. (t) r > = U (t)> r (t) = U(t)U, where (t) r > and r (t) are a state ket and a density operator in a new frame. Liouville-von Neuman equation for the density operator in the frame given by (eqs ) d r (t)/dt = i[ r (t), H e ] with H e = UHU iu{du /dt} [.66] r (t) = exp(-ih e t) r ()exp(ih e t) U for a Rotation Transformation A rotation transformation along the z ax by an angle φ given by R z z(φ) = exp(-iφi φ z ) Transformation to a frame rotating at an angular veloity ω rf given by U = R z (-ω rf t) = exp(i ω rf I z t) Then, H e = UHU iu{du /dt} = exp(iω rf I z t)hexp(-iω rf I z t) - ω rf I z, where iu{du /dt} = - ω H rf I z What e H e when H= I Z? = exp(iω rf I z t) I Z exp(-iω rf I z t) ω rf I z = I Z ω rf I z 13

14 RF Hamiltonian in the rotating frame U = exp(-iω rf I z t) H e = UHU iu{du /dt} = exp(iω rf I z t)hexp(-iω rf I z t) - ω rf I z, H = -μ B = ω I z + ω 1 I x os(ω rf t) = ω I z +ω 1 R Z (ω rf t)i X R Z (ω rf t) + ω 1 R Z (ω rf t)i X R Z (ω rf t). ~ ω I z + ω 1 U I XU exp(iω rf I z t)hexp(-iω rf I z t) = ω UI z U + ω 1 UU I X UU = [Q1] + [Q] H e = ω I z + ω 1 I X - ω rf I z RF Hamiltonian in the rotating frame H e = UHU iu{du /dt} = exp(iω rf I z t)hexp(-iω rf I z t) - ω rf I z, H = -μ B = ω I z + ω 1 I x os(ω rf t+φ) ~ ω I z +ω 1 R Z (ω rf t+φ)i X R Z (ω rf t+φ). exp(iω rf I z t)hexp(-iω rf I z t) = Q1 H e = ω I z + ω 1 R Z (φ)i X R Z (φ) ω rf I z 14

15 RF Hamiltonian & Rotating Frame Spin Hamiltonian in a stati magneti field and a transverse RF field given by H = -μ B B = ω I z + ω 1 I x os(ω RF t+φ) = ω I z +ω 1 {I x os(ω RF t+φ)-i Y sin(ω RF t+φ)} +ω 1 {I x os(ω RF t+φ) + I Y sin(ω RF t+φ)} ~ ω I z +ω 1 R Z (ω RF t+φ)i X R Z (ω RF t+φ). In the rotating frame, H e = UHU iu{du /dt} = (ω -ω rf )I z + ω 1 {osφ I X +sinφ I Y }. RF Hamiltonian & Rotating Frame Spin Hamiltonian in a stati magneti field and a transverse RF field given by H = -μ B B = ω I z + ω 1 I x os(ω RF t+φ) = ω I z +ω 1 {I x os(ω RF t+φ)-i Y sin(ω RF t+φ)} +ω 1 {I x os(ω RF t+φ) + I Y sin(ω RF t+φ)} ~ ω I z +ω 1 R Z (ω RF t+φ)i X R Z (ω RF t+φ). In the rotating frame, H e = UHU iu{du /dt} = (ω -ω rf )I z + ω 1 {osφ I X +sinφ I Y }. 15

16 The effet of an RF field in the Rotating Frame H e = (ω -ω rf )I z + ω 1 (osφ I X + sinφ I Y ). σ e (t) = [Q1] σ e ()[Q] On-resonane ase: (ω -ω rf ) << ω 1 When φ =, H ~ ω 1 I X σ e (t) = exp(-iω 1 I X t)(ai z )exp(iω 1 I X t) = a(i Z os ω 1 t - I Y sin ω 1 t). Q. What does th mean? Q. When t = π/ω 1, σ e (t)= -I [Q1] Y When t = π/ω 1, σ e (t)= -I [Q] Z z Rotation Iz osα - Iy sinα y x p Rotation along x 16

17 Observing a signal in the rotating frame σ e (t) = exp{-ii z t}i x exp {ii z t} = I x ost + I Y sin t, where = ω ω rf. M + = M x + im y = γ(i x + ii y ) <M + > e = Tr(M + σ e (t)) = γtr{(i X + ii Y )(I x os t + I Y sin t)} = γ{os [Q1] t + in t}/, where we used Tr(I x I y ) = and Tr(I x ) = Tr(I y ) = 1/ The effet of an general RF field H e = (ω -ω rf )I z + ω 1 (osφ I X + sinφ I Y ). = n x I X +n Y I Y +n Z I Z. σ e (t) = [Q1] σ e ()[Q] 17