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Lecture #2: NMR Operators, Populations, and Coherences

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Lecture #2: NMR Operators, Populations, and Coherences Lecture #2: NMR Operators, Populations, and Coherences • Topics – Quantum description of NMR – The Density Operator – NMR in Liouville Space – Populations and coherences • Handouts and Reading assignments – Kowalewski, Chapter 1 – van de Ven, appendices A-C 1 Why Quantum Mechanics for NMR? • Classical picture of NMR (i.e. Bloch equations) is valid for collections of non-interacting spins. • Individual spins don’t have an associated T1 or T2. • NMR relaxation is described statistically and is driven by interactions among spins, e.g. – Nuclear dipolar and J couplings – Chemical exchange – Electron-nuclear interactions: shielding, dipolar coupling, J coupling • Such interactions are best described using semi-classical or fully quantum mechanical descriptions. 2 Wave Functions • For the classical concept of a trajectory (succession in time of the state of a classical particle), we substitute the concept of the ! quantum state of a particle characterized by a wave function, ψ ( r , t ) . ! • ψ ( r , t ) - contains all info possible to obtain about the particle € - interpreted as a probability amplitude of the particle’s presence € with the probability density given by: ! ! 2 ! dΡ(r ,t) = Cψ(r ,t) d 3r , C constant. ! 1 ! 2 ! ∫ dΡ(r ,t) =1 = ∫ ψ(r ,t) d 3r << ∞ C € square-integrable! ! 2 ! - wave functions typically normalized, i.e. ∫ ψ(r ,t) d 3r =1 3 € € € QM Math: Kets and Bras • Important to be familiar with the concept of treating functions as elements in a vector space. • Example: consider f(n) for 1 ≤ n ≤ N, n integer. " f % f 1 3 ! $ ' f2 fN f2 f1 f f = f(n) 4 $ ' ... $ " ' n f # N & ! • Each wave function ψ ( r ) is associated with an element of a linear vector space F and is denoted as ψ (called a “ket”). Dirac ! € notation ψ (r ) ⇔ ψ ∈ F € “ ” • Elements of the dual space of€ F are denoted as χ (called a bra ). • A linear vector€ space with a defined metric, in this case the scalar product χ ψ , is called a Hilbert Space. € 4 QM Math: Bases and Operators n • We can express ψ as a linear combination of other kets: ψ = ∑ci fi i=1 N • If it follows that: ∑ci f i = 0 implies ci = 0 for all i , then the kets i=1 fi are called linearly independent. • In an N-dimensional vector space, N linearly independent kets € constitute a basis, and every element ψ in F can be written as n € ψ = ψ f ∑ i i where ψi is the coefficient of the ket f i . i=1 $" 1 if i = j f f • We typically deal with orthonormal€ basis: i j = # %$ 0 if i ≠ j € € • An operator, O ˆ , generates one ket from another: Oˆ ψ = ξ Example of a linear operator: Oˆ ( λ ψ + µ ξ ) = λOˆ ψ + µOˆ ξ scalars € € 5 € QM Math: Vectors and Matrices ψ • When a ket, , is expressed as a linear combination! of basis kets, fi ψ , then a corresponding column vector, , can be constructed. # ψ1 & % ( N ! ψ ! € % 2 ( † ∗, ∗, , ∗ ψ = ∑ψi f i ψ = ψ ψ = ψ1 ψ2 … ψn ( ) € i 1 % " ( € = % ( $ψ n ' • Similarly, operators have matrix representations. ! ! € € € x Oˆ y where O is a n x n matrix with elements: ˆ = x = Oy Oij = f i O f j € z # x0 & # 0 & z ! % ( ! % 2 2 ( Remember: vector and 0 y x y ψ ψ = % 0 ( ψ = % 0 + 0 ( €matrix representations€ % ( % ( € depends on the basis set! € $ z0 ' $ z0 ' € y x 0 x " 0 y basis:{ x , y , z } { x " , y " , z } x y "€ € 6 € € € € € € QM Math: Eigenkets and Eigenvalues • If Oˆ ψ = λ ψ scalar then ψ is called an eigenket of O ˆ with eigenvalue λ. € • The trace of a matrix is defined at the sum of the diagonal elements: € € Tr(O) = ∑Oii i • Since the trace of a matrix is invariant under a change of basis, we can also talk about the trace of the corresponding operator. € Tr Oˆ = λ ˆ ( ) ∑ i where λi are the eigenvalues of O i 7 € € QM Math: Liouville Space • Operators defined on an n-dimensional Hilbert space, are themselves elements of an n2-dimensional vector space known as Liouville space. • The trace is the metric in Liouville space: ( Aˆ | Bˆ ) = Tr ( Aˆ † B ˆ ) , and in contrast to Hilbert space, the product of two Liouville Space elements is defined. a Liouville space Aˆ , Bˆ ∈ L Aˆ B ˆ = Cˆ ∈ L In general, AˆBˆ ≠ BˆAˆ. • Operators that work on elements of a Liouville space are called superoperators (denoted with a double hat). € ˆ ˆ ˆ € AO 1 = O2 , and we will deal with just one superoperator: the ˆ commutator: Aˆ B ˆ ≡ Aˆ , Bˆ = Aˆ B ˆ − Bˆ Aˆ Do superoperators have [ ] associated eigenoperators? € What about supermatrices? 8 € Summary of Vector Spaces Hilbert Space H NMR can be described {ψi },i = 1,..., n in Hilbert space ... Liouville Space L …or in Liouville (operator algebra) space. ˆ 2 Oi ,i = 1,..., n { } Key Concept NMR is easiest to Superoperator Algebra S understand in ! ˆ $ 4 Liouville space! " Sˆ % ,i = 1,..., n # i & See R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, 1990. 9 Postulates of QM • The state of a physical system is defined by a ket ψ ( t ) that belongs to a linear vector space known as a Hilbert Space. • Every measurable quantity A is described by an Hermitian operator A ˆ whose eigenkets form a basis in state space. • The only possible result of the measurement of A is an eigenvalue ˆ of the A , the probability of obtaining eigenvalue an is 2 P(an ) = un ψ where u n is the associated eigenket, and the state of the system immediately after the measurement is ψ = un . • The time evolution of the ψ ( t ) is governed by the Schrödinger € € equation: ∂ ψ(t) = −iHˆ (t)ψ(t) ∂t where H ˆ ( t ) , known as the Hamiltonian, is the operator for the observable H(t) associated with the total energy of the system. € 10 Spin, Angular Momentum, and Magnetic Moment • Spin, angular momentum, and magnetic moment operators are linearly related. ˆ ˆ µˆ p = γL p = γ!I p , p = {x,y,z} magnetic moment angular momentum spin with the following commutators: Iˆ ,Iˆ iIˆ Iˆ ,Iˆ iIˆ Iˆ ,Iˆ iIˆ [ x y ] = z [ y z ] = x [ z x ] = y € 1 1 • For a spin ½ particle: Iˆ + = + + and Iˆ − = − − z 2 z 2 ˆ eigenkets of I z € € € • Matrix representation in { + , − } basis: € 1" 0 1% 1# 0 −i& 1# 1 0 & I x, = $ ' I y, = % ( I z, = % ( 2# 1 0& 2$ i 0 ' 2$ 0 −1' The Pauli matrices 11 € € € Isolated Spin in a Magnetic Field • Goal: Find the appropriate wavefunction ψ ( t ) that describes a system consisting of a nucleus (spin = ½) in a uniform magnetic field. • Procedure: € ∂ – Given: Schrödinger’s Equation: ψ(t) = −iHˆ (t)ψ(t) ∂t – Find Hˆ (t). – Solve for ψ . € – Compute quantities of interest: e.g. components of magnetic moment µˆx , µˆy , and µˆz . € Previously showed these correspond to the familiar quantities Mx, My, and Mz. 12 A Collection of Spins • In a typical experiment, the number of nuclear spins, N, can be very big, e.g. 1024, and the complete quantum state is described by the wavefunction (or state vector): Rather unwieldly N for large N! ψ = ∑cn ψn n=1 • To make matters worse, we rarely, if ever, know ψ precisely, e.g. cn may be different for each spin. At best, we have only a statistical model for the state of the system. € 13 Density Operator for a Pure State • Consider a spin system in a pure state with normalized state vector ψ(t) = ∑ci (t) ui where { u i } form an orthonormal basis. i • We previously showed that we could define an operator: σˆψ (t) = ψ(t) ψ(t) for which € € - Expectation of observable Aˆ : Aˆ (t) u ˆ (t) u u Aˆ u Tr ˆ (t)Aˆ = ∑ j σ ψ i i j = {σ ψ } i, j - Time evolution: ∂ ˆ ∂ ! # σ ψ (t) = " ψ(t) ψ(t) $ ∂t ∂t = −i[Hˆ , ψ(t) ψ(t) ] ˆ = −iHσˆψ 14 € Density Operator: Statistical Mixture • Consider a system consisting of a statistical mixture of states ψ n ˆ with associated probabilities pn, and ai be an eigenvalue of A with associated eigenket ui . • While there is no “average state vector”, it turns out there is an “average operator” known as the density operator. ui ui a = p a p Tr ˆ Pˆ … in general: P( i ) ∑ nPn ( i )= ∑ n {σ n i } n n " € % ˆ ˆ ˆ ˆ = Tr#∑ pnσ nPi & = Tr{σ Pi } $ n ' ˆ ˆ where σ = ∑ p n σ n is, by definition, the density operator for the system. n ∂ ˆ • It is easy to show that: Aˆ = Tr σˆ Aˆ and σˆ = −iHˆσˆ { } ∂t Liouville-von Neumann ensemble average equation 15 System of independent spin= ½ nuclei ˆ • Most convenient basis set is the eigenkets of H 0 : { + , − } + − $ 2 * ' c c c + " % L - “longitudinal magnetization” & + + − ) L I σ = $ ' I - “transverse magnetization” & * 2 ) € −€ c c c # I L & “vector” in a 4D % + − − ( € Liouville space called € “ ” • σ for a one-spin system can be expressed as: coherence space € " L1 % $ ' # 1 0& # 0 1& # 0 0& # 0 0& I $ 1 ' σ = L1% ( + I1% €( + I2% ( + L2% ( σˆ € $ 0 0' $ 0 0' $ 1 0' $ 0 1' $ I2 ' € ˆ ˆ ˆ ˆ $ ' In operator form: σˆ = L1T11 + I1T12 + I2T21 + L2T22 # L2 & 1 i 1 # 1 0& # 0 2& # 0 − 2& € # 2 0 & • or … σ = a1% ( + a2% ( + a3% ( + a4 % ( € 1 i 1 Iˆ $ 0 1' $ 2 0' $ 2 0 ' $ 0 − 2' z € € Iˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ y σˆ = a1E + a2I x + a3I y + a4 I z E ,I x,I y,I z { } A 3D subspace of σˆ “product operator” basis set coherence space Iˆ 16 € x € € Coupled Two-Spin System ˆ • Eigenkets of H0 : { ++ , + − , −+ , − − } ++ + − −+ − − Off-diagonal elements called “coherences” ++ ! P1 C1,2 C1,3 C1,4 $ + − # C P C C & σ 2,1 2 2,3 2,4 −+ # C3,1 C3,2 P3 C3,4 & − − # & " C4,1 C4,2 C4,3 P4 % Diagonal elements called “populations” € • As with the single-spin case, the product operators form a convenient orthonormal basis set (16 in total).
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