Manifestations of classical physics in the quantum evolution of correlated spin states in pulsed NMR experiments. Abstract

Transcription

1 Manifestations of classical phsics in the quantum evolution of correlated spin states in pulsed NMR eperiments Martin Ligare Department of Phsics & Astronom, ucknell Universit, Lewisburg, PA 7837 Dated: August, 07) Abstract Multiple-pulse NMR eperiments are a powerful tool for the investigation of molecules with coupled nuclear spins. The product operator formalism provides a wa to understand the quantum evolution of an ensemble of weakl-coupled spins in such eperiments using some of the more intuitive concepts of classical phsics and semi-classical vector representations. In this paper I present a new wa in which to interpret the quantum evolution of an ensemble of spins. I recast the quantum problem in terms of mitures of pure states of two spins whose epectation values evolve identicall to those of classical moments. Pictorial representations of these classicall evolving states provide a wa to calculate the time evolution of ensembles of weakl-coupled spins without the full machiner of quantum mechanics, offering insight to anone who understands precession of magnetic moments in magnetic fields. PACS numbers: Electronic address: mligare@bucknell.edu

2 I. INTRODUCTION The past several decades have seen an eplosion in the development of new and powerful eperimental techniques investigating coupled nuclei using multiple-pulse NMR eperiments. The sophistication of these techniques presents a pedagogical problem: How can these techniques be introduced to students and practitioners who ma never have the opportunit to develop a deep understanding of the quantum mechanical formalism that provides a quantitative description of all aspects of the dnamics of coupled spins? In this paper I identif classicall-evolving states in a full quantum mechanical treatment of the time evolution of coupled spins in multiple-pulse NMR eperiments. Pictorial representations of these quantum states have the potential to provide phsical insight to anone who understands classical precession of moments in magnetic fields. The challenge of reconciling quantum and classical pictures of magnetic resonance goes back to the earl das of NMR. Purcell at Harvard) and loch at Stanford) had ver different perspectives on NMR differences so great that their respective achievements were termed the two discoveries of NMR in an ecellent review reconciling the quantum and classical points of view of single-spin NMR []. It is now understood that classical phsics, with accompaning pictorial representations of vector moments, is sufficient to describe and understand the time evolution of quantum epectation values, and thus the results of all NMR eperiments involving ensembles of single spins [, 3]. The correspondence between classical and quantum results for precessing spins has also been discussed in terms of Ehrenfest s theorem in several undergraduate tets [4, 5]. Identifing classical descriptions and representations of NMR eperiments on coupled spins has proved a more difficult challenge. Quantum mechanics provides, of course, a complete description of all eperiments, but quantum mechanics is not intuitive for those not well versed in the formalism. The product operator method was introduced in 983 [6 8] with the epress intent of providing a middle course between the full quantum mechanical densit-matri theor and the more intuitive classical and semiclasical vector models. In the ensuing ears man authors have proposed a variet of pictorial models to help students navigate this middle course between classical and quantum mechanical perspectives see, for eample, [9 4]). In this paper I present a new middle course. Like the product operator formalism, m course is full quantum mechanical, but it is based on the identification of pure quantum

3 states of two spins that evolve in was that have eact analogues in the evolution of classical magnetic moments. As I show, the terms in the product operator epansion of the densit matri which do not evolve analogousl to classical moments) can all be associated with mitures of two of these classicall evolving pure quantum states. The pure quantum states I introduce are eas to represent pictoriall, and these representations provide an eas wa to determine the eact densit matri for ensembles of weakl-coupled spins in pulsed NMR eperiments. The formalism I introduce is not intended to replace product operators as a tool for analing eperiments on coupled spins. Rather, it is intended to provide a new perspective, informed b classical phsics, on the evolution of the terms in the product operator epansion of the densit matri. The tools discussed in this paper ma also find use in introducing multiple-pulse NMR eperiments to those without a strong background in quantum mechanics. II. CLASSICALLY EVOLVING QUANTUM STATES OF SINGLE SPINS A. Pictorial representations of general single spin states In this section I consider individual spin- nuclei. The nuclei have a magnetic moment m which is proportional to the angular momentum, i.e., m = γi, where I is the nuclear angular momentum vector and γ is the gromagnetic ratio. In a constant magnetic field,, the moments eperience a torque τ = m, ) and precess around the ais of the field; if the field points in the + direction in some coordinate sstem, the moments precess in a clockwise sense when looking down from the + direction), and themagnitudeoftheangular frequenc is ω 0 = γ, so that theaimuthal angle of the moment s direction is given b φ = φ 0 ω 0 t. The energ associated with the orientation of a moment is E = m. ) In quantum discussions of magnetic resonance, it is conventional to use as a basis the spin-up and spin-down states + and often labeled as α and β respectivel). When the magnetic field points along the -ais, these states are eigenstates of the operator 3

4 Î corresponding to the observable -component of angular momentum, with eigenvalues / and / respectivel. Quantum angular momentum operators will be defined without the factor.) These states are simultaneousl eigenstates of the Hamiltonian with energ eigenvalues ω 0 / and + ω 0 /. I will use this conventional basis for all matri representations of operators and states in this paper. Operators will be designated with a hat, and matri representations of operators without one.) Outcomes of measurements made on spins are governed b the rules of quantum mechanics, and therefore no classical pictorial representation can completel represent all aspects of a spin state. On their own, the basis states ± do not displa man aspects of the general classical behavior of spins. Measurements of the -component of angular momentum alwas give the same value, but measurements of an component in the - plane give random results of ± /. There is no sense in which moments in these basis states ehibit precession. States with behavior that corresponds more closel to classical epectations, however, are well known and eas to construct. These states are discussed in man magnetic resonance tets and papers see, for eample [, 3, 5]), and the general mapping of the dnamics of two-state quantum sstems to the geometr of 3-space was elucidated b Fenman, et al. [6]. I will briefl summarie the results that can be found in the cited tets as well as man other tets and articles). Consider a magnetic moment in the linear-combination state θ,φ = cos θ + +eiφ sin θ cos θ e iφ sin θ, 3) where θ and φ correspond to conventional angular spherical coordinates in real 3-space. Measurement of the component of angular momentum along an ais defined b the angles θ and φ will result in a value + / with a probabilit of. In other words, this is state is an eigenstate of the operator Î θ,φ = sinθcosφ Î +sinθsinφ Î +cosθ Î. 4) The state θ,φ can be represented pictoriall b an arrow along the direction given b the angles θ and φ, as in Fig.. In this arrow representation, the spin-up state + is represented b an arrow pointing in the + direction, and the spin-down state b an 4

5 θ θ,φ φ FIG. : Arrow representation of the spin state θ;φ = cos θ + +eiφ sin θ. Measurement of the component of the angular momentum along the direction of the arrow gives the value + / with probabilit. arrow pointing in the direction. The state with spin-up along the ais, given b the linear combination of the + and states + = θ=π/,φ=0 = + + ), 5) is represented b an arrow along the +-ais, and the general state given in Eq. 3) is represented b an arrow in the direction given b θ and φ. The results of measurements of angular momentum components that don t lie along the ais of an arrow can not be predicted b taking classical geometric projections of the arrow ontootherdirections; suchmeasurementswillalwasieldrandomvaluesof+ /and /. The arrows displa information about the probabilities of obtaining + / or / for an measurement, but the can not be regarded as classical vectors when predicting results of individual spin measurements. The arrows do, however, evolve in time identicall to classical magnetic moment vectors, as I will discuss in the net section. It should be noted that the arrows used in this representation are not the same as the arrows representing magnetiation vectors in semi-classical representations of magnetic resonance, like the well-known loch vectors. As a simple eample, consider a 50:50 miture of spin-up and spin-down nuclei. Although the magnetiation vectors of the two spin states cancel, the densit matri of the not ero. The information about the densit matri is retained in representation with two arrows, with appropriate weighting factors. In addition, 5

6 the arrows in the arrow representation do not decompose into components along the aes.. Evolution of single-spin states in magnetic fields The advantage of the arrow representation of quantum spin states becomes apparent when the spin is placed in a magnetic field. The quantum spin state evolves in such a wa that the arrow representing the state precesses eactl as a classical magnetic moment with the same orientation. For eample, in a constant magnetic field in the + direction, the arrow representing the state + = θ=π/, φ=0 and pointing in the + direction) will precess in the - plane as the state evolves into θ=π/, φ= γt. In contrast, the spin-up state +, represented b a arrow pointing along the direction of the field, is a stationar quantum state and does not precess, just as a classical moment aligned with a field does not precess. Classical phsics provides a complete description of the evolution of individual quantum spin states; it also describes the time-dependent quantum epectation values of observables for ensembles of spins. It is onl when measurements made on individual spins are considered that classical and quantum predictions differ. III. REPRESENTING PURE QUANTUM STATES OF SPIN PAIRS A. asis states for weakl coupled spin pairs For weakl interacting inequivalent spins in an isotropic liquid phase, in a magnetic field with magnitude 0 pointing in + direction, the approimate Hamiltonian is [] Ĥ ω 0 Î ω 0 Î +π J ÎÎ, 6) whereω 0 = γ 0,ω 0 = γ 0,andtheconstantscalarJ characteriesstrengthofthecoupling between the spins. The conventional basis for two spins consists of the four states α,α = +;+, α,β = +;, β, α = ; +, and β, β =,. These states are eigenstates of the 6

7 +; + +; + +; θ=π/,φ φ FIG. : Eamples of double-arrow representations of states of two correlated spins. The arrows give directions for which simultaneous measurements of angular momentum components will both ield + / with probabilit. Hamiltonian, with energies E +,+ = ω 0 ω 0 +πj ) 7a) E +, = ω 0 +ω 0 πj ) 7b) E,+ = ω 0 ω 0 πj ) 7c) E, = ω 0 +ω 0 +πj ). 7d) As in the case of a single spin, these energ eigenstates states do not ehibit precession around the fields in the -direction.. Pictorial representation of pure states of spin pairs In this section I etend the arrow representation appropriate for single spins and introduce a double-arrow representation to cover cases of two correlated spins. These states are direct products of two single-spin states like those detailed in Eq. 3). As a first eample, consider the two-spin states represented in Fig.. The arrows in this figure represent the two directions for which the simultaneous measurement of the component of angular momentum for a single spin are both certain to ield + /. For the two-spin state on the left, simultaneous measurement of the -component of angular momentum of spin and -component of angular momentum spin are both certain to ield values of + /; I label this state +; The two-spin state in the middle ields a similar result for measurement of the -component of spin and the -component for spin : both component measurements are certain to ield values of + /. The double-arrow states are 7

8 not limited to components along Cartesian aes. An eample in which one of the angular momentum components is measured along an arbitrar ais in the - plane is illustrated on the right in Fig.. These double-arrow states can be written in terms of the standard basis of projections along the -ais. For eample, the state on the left in Fig. can be written +;+ + + = + + ) + = +;+ + ;+ ) 0. 8) 0 More generall, two-spin states can be written θ A,φ A ;θ,φ θ A,φ A A θ,φ = cos θ A + A +e iφ A sin θ ) A A cos θ + +e iφ sin θ ) = cos θ A cos θ ) +;+ + e iφ cos θ A sin θ ) +; + e iφ A sin θ A cos θ ) ;+ + e iφ A+φ ) sin θ A sin θ ) ; cos θ A cos θ e iφ cos θ A sin θ. 9) e iφ A sin θ A cos θ e iφ A+φ ) sin θ A sin θ IV. TIME EVOLUTION OF DOULE-ARROW STATES The time evolution of double-arrow pure states is straightforward to deduce from the simple time dependence of the eigenstates of the Hamiltonian given in Eq. 6); the details for a general double-arrow state are given in the Appendi. In some special cases the quantum double-arrow states remain in a factored form as the product of two spin states that both evolve in a simple manner. These special case states are the subject of this section, 8

9 and will be the states used in analsis of a multi-pulse eperiment. It should be noted that the classical analogs of these special-case quantum states also ehibit simple time evolution. Conversel, the quantum states that don t factor in this manner correspond to those classical configurations with coupling that leads complicated time evolution. A. Evolution of uncoupled spin pairs in magnetic fields For uncoupled spins, the quantum mechanical time evolution of the double-arrow states in constant magnetic fields is eas to visualie, because each of the single-arrow states making up the double-arrow evolves classicall and independentl. The factoring of these quantum states is illustrated in the Appendi.) For eample, in a constant field in the + direction with magnitude 0, the state +,+ illustrated in Fig. precesses clock-wise as viewed from the +-ais) at a rate ω 0 = γ 0, evolving into another double-arrow state: ψ0) = +;+ ψt) = θ =π/,φ = ωt; 0 +, 0) with the tip of the arrow corresponding to spin tracing out a circle in the - plane. In a frame rotating clockwise around the ais with angular frequenc ω ref, the tip of the arrow precesses at the offset frequenc Ω 0 = ω 0 ω ref. For the state in the middle in Fig., each of the arrows precesses at its own rate, giving in the rotating frame) ψ0) = +;+ ψt) = θ =π/,φ =π/ Ω 0 t; θ =π/,φ = Ω 0 t, ) The same kind of evolution also occurs during strong rf pulses, as is illustrated in Fig. 3. In the hard pulse approimation, a π/) pulse applied to the state +; rotates the arrow for spin to the -ais, while leaving the arrow for spin pointing in the + direction, resulting in the spin-pair state +; +.. Evolution of double-arrow states of coupled spin pairs in a magnetic field For weakl-coupled spins pairs the time evolution of a sstem starting in a double-arrow state is in general more comple than the simple precession discussed in the previous section, but it simplifies if one of the spins is aligned or anti-aligned) with the eternal magnetic field. This is true for classical spins as well, because an aligned moment doesn t precess, 9

10 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 3: The effect of a π/) pulse on a double-arrow state of two spins. +; + Ω 0 +πj +; Ω 0 πj FIG. 4: Time evolution in the rotating frame) of a double-arrow state of two weakl-coupled spins in an eternal field. The precession rate of moment depends on the orientation of moment. resulting in a constant field due to the moment.) When one of the arrows is aligned with the field, the other arrow simpl precesses around the field, and the spin pair remains in a double-arrow state as it evolves. The angular frequenc of the precessing moment depends on the orientation of the stationar moment with respect to the field. An eample of this is illustrated in Fig. 4. A more detailed discussion of this orientation dependence is contained in the Appendi. 0

11 V. DOULE-ARROW STATES AND PRODUCT OPERATORS A. Product operators as mitures of pure states The double-arrow states I have introduced have a close relationship with what are known as product operators [6 8]. The product operator formalism provides a decomposition of the densit matri of an ensemble of spins into terms that are orthogonal with respect to the trace), and that evolve in eas-to-characterie was [7]. The terms in a product operator epansion are not densit matrices of pure states. As I will demonstrate, however, each of the terms in the product operator epansion is related to the densit matri of a 50:50 miture of two classicall evolving double-arrow states. The correspondences between representative product operators and mitures of spin-pair double-arrow states are illustrated in Fig. 5, and will be discussed below. As an eample of how this association works, consider the term in the epansion of the densit matri labeled I. This term corresponds to the densit matri I = ) Anon-eroI termimpliesanetpolariationofnucleusalongthe-ais, withnopreferred orientation for nucleus. The phrase no preferred orientation is an inherentl classical epression; the spins labeled must be in some quantum state, and all pure quantum spin states ehibit a polariation in some direction. The phrase no preferred direction implies that the ensemble of spins is in a miture of states, and the miture must be consistent with the isotropic distribution of results for measurements of the components of angular momentum of spin. A 50:50 miture of two appropriatel chosen double-arrow states satisfies these criteria. As one eample, consider a 50:50 miture of the double-arrow states +;+ i and +; i, 3)

12 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 5: Mitures of double-arrow states that correspond to representative product operators. The solid blue lines represent a single pure quantum state of a spin pair, and the dotted red lines represents a different single pure state. with associated densit matrices i 0 0 ρ +,+ = i i 0 0 and ρ +, = i ) The densit matri for a 50:50 miture of these two pure states is ρ +,+ + ρ +, = ) The epectation value for an component of angular momentum of nucleus is ero, i.e.,

13 TrρI j ) = 0, for an component j. This means that there is no preferred orientation for nucleus in the ensemble of spin states. The epectation value of the -component of angular momentum of nucleus is / TrρI ) = /, as desired. Comparing Eqs. ) and 5) gives the relationship between the product operator I and the densit matri for a miture of the two double-arrow pure states: ρ +,+ 4 ) E + ρ +, 4 ) E = I, 6) where E is the identit matri. The product operator is simpl twice the densit matri of a 50:50 miture of double-arrow states, with a term proportional to the identif matri subtracted to make the product operator miture traceless. Similar considerations give the general results ρ +j,+k 4 ) E + ρ +j, k 4 ) E = I j 7a) ρ +k,+j 4 ) E + ρ k,+j 4 ) E = I j. 7b) where j and k indicate directions in space. In words, a single-spin product operator is associated with a miture of two double-arrow states, with one arrow from each of the states pointing along the direction specified b the product operator, and the other arrows opposite to each other along an arbitrar direction. Note that the choice to use -components for spin in the 50:50 miture of Eq. 6) is at this point arbitrar; components along an opposing directions would work just as well, giving the same result for the densit matri. Similar arbitrariness will arise in the representation of other terms in the product operator epansion too. In Section VII, where I discuss a multiple pulse eperiment, I will make choices from the set the possible representations that make the time evolution the simplest, matching that epected for similarl aligned classical moments. As an eample of a two-spin product operator, consider the product operator labeled I I, corresponding to the densit matri i I I = 0 0 i 0. 8) 0 i 0 0 i

14 Adensitmatriepansionwithanon-eroI I termimpliesthatmeasurementsofthe -component of spin and the -component of spin will be correlated, i.e., a measurement ielding a positive value of the -component of angular momentum of nucleus is more likel than not to be accompanied b a positive -component of the angular momentum of nucleus, and measurement of a negative value of the -component of angular momentum of nucleus is more likel than not to be accompanied b a negative -component of the angular momentum of nucleus. Consider the 50:50 miture of the double-arrow states in which the correlation of the and components of the spins is manifest: +;+ i i and ; i, 9) with associated densit matrices i i ρ +,+ = i i 4 i i i i i i and ρ, = i i. 0) 4 i i i i The densit matri for a 50:50 miture of these two pure states is ρ +,+ + ρ, = i 0 i 0. ) 0 i 0 i 0 0 ComparingEqs.8) and)givestherelationshipbetweentheproductoperatori I and the densit matri for a miture of two double-arrow pure states: ρ +,+ 4 ) E + ρ, 4 ) E = I I ). ) Once again, the product operator is simpl twice the densit matri of a miture of smmetricall-oriented double-arrow states with a term proportional to the identit matri subtracted). 4

15 More generall, for a two-spin product operator term we have ρ +j,+k 4 ) E + ρ j, k 4 ) E = I ji k ), 3) where j and k label directions in space. In words, a two-spin product operator is associated with a miture of two double arrow states; in one of the double arrow states the directions are those specified in the labeling of the product operator, and in the other the directions of the arrows are reversed. Additional eamples using the generalied rules associating product operators and doublearrow mitures are illustrated in Fig. 5.. Component decomposition of product operator mitures The arrows in the double-arrow representation of pure states should not be confused with arrows representing vectors in semi-classical models; the are simpl indicators of a direction. One wa in which the are clearl not vectors is that the do not decompose into components as classical vectors do. A naive decomposition into components of the double-arrow state +;θ=π/,φ depicted on the right in Fig. ) doesn t work, i.e., +;θ=π/,φ cosφ +;+ +sinφ +;+. 4) In contrast, the densit matrices of some mitures of smmetricall-oriented double-arrow states can be broken up in a component-like manner. A detailed discussion is contained in the Appendi.) The mitures that work are those that are associated with product operators. Two eamples are illustrated in Fig. 6. If we combine the densit matri for the state +;θ=π/,φ with that of the state ;θ=π/,φ, as shown in the top row of the figure with subtraction of appropriate terms proportional to the identit matri to make them traceless), we find ρ +,φ E ) + ρ,φ E ) [ = cosφ ρ +,+ E ) + ρ,+ E )] [ +sinφ ρ +,+ E ) + ρ,+ E )] 4 4 = cosφi +sinφi. 5) Combining the densit matri for the state +;θ=π/,π/ φ with that of the state 5

16 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 6: Geometric decomposition of mitures of smmetricall oriented double-arrow states. Weighting factors for the terms in the decomposition are given b Eqs. 5) and 6). ;θ=π/,π/+φ as shown in the bottom row of the figure with subtraction of appropriate terms proportional to the identit matri to make them traceless) we find ρ +,π/ φ E ) + ρ,π/+φ E ) [ = cosφ ρ +,+ E ) + ρ,+ E )] [ +sinφ ρ +,+ E ) + ρ, E )] 4 4 = cosφi +sinφi I. 6) The trigonometric factors in the preceding equations are not reflected in the length of an arrows in the double-arrow representation of pure states. Rather, the trigonometric factors are simpl the weights for the terms in the densit matri when it is rewritten in terms of pure states with arrows along the coordinate aes. 6

17 VI. TIME EVOLUTION OF PRODUCT OPERATOR MIXTURES VIA PICTO- RIAL REPRESENTATIONS In the preceding section I related terms in the product operator epansion with mitures of two classicall evolving double-arrow states. To visualie the time evolution of the ensemble population corresponding to a product operator term it is onl necessar to consider two of the evolving double arrows. A. Evolution due to eternal fields The time evolution of 50:50 mitures of representative double-arrow states in eternal magnetic fields are illustrated in Fig. 7. The illustrations are drawn in the frame rotating at angular frequenc ω ref, and the offset frequencies are given b Ω 0 = ω 0 ω ref and Ω 0 = ω 0 ω ref. The initial states are on the left, and the states after one quarter of a relevant precession period are illustrated on the right. The associated product operators are also indicated. In the illustrated evolutions, all evolution is due to the eternal field, and spinspin coupling is assumed to be negligible. In all cases an arrow along the ais of the field is stationar, and an arrow in a plane perpendicular to the ais of the field ehibits simple precession. Notice that the eternal field causes single-spin product operators to evolve into new single-spin operators or combinations of single-spin operators), and two-spin operators to evolve into two-spin operators or combinations of two-spin operators). Field driven evolution of all terms in the product operator epansion can be understood with similar pictures. Not onl do the pictures Fig. 7 present a ver classical picture of precessing spins, the also provide simple tools for determining the full quantum densit matri of the sstem. Epressions for each of the quantum states represented b the double arrows are given b Eq. 9), from which the densit matri is eas to calculate.. J-coupling evolution The time evolution of 50:50 mitures of representative double-arrow states due to J- coupling of the spins is illustrated in Fig. 8. The initial states are on the left, and the states after one quarter of a relevant precession period are illustrated on the right. The associated 7

18 I t = π/ω 0 I Miture of +;+ and +; = 0 ˆk: Ω 0 I t = π/ω 0 I Miture of +;+ and +; = 0 ˆk: Ω 0 Ω 0 I I t = π/ω I I Miture of +;+ and ; = î: Ω ω I I t = π/ω 0 cosω 0 I I sinω 0 ti I Miture of +;+ and ; = 0 ˆk: Ω 0 Ω 0 Ω 0 Ω 0 Ω 0 t Ω 0 t FIG. 7: Time evolution of mitures of double-arrow states of spin pairs in eternal magnetic fields. Initial states are on the left, and states after one quarter of a precession period are on the right. Illustrations are drawn in a rotating frame. product operators are also indicated. The illustrated evolutions are drawn in the appropriate rotating frame so that the onl include the contribution from the spin-spin coupling. In the top illustration, the J-coupling causes the spin coupled to the spin-up spin solid blue double-arrow) to decrease it s precession rate, and the spin coupled to the spindown spin red dotted-line double-arrow) to increase it s rate. This is an eample of a 8

19 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 8: Time evolution of mitures of double-arrow states of spin-pairs due to J-coupling of the spin pairs. The top figure is drawn in a frame rotating at ω 0, and the middle figure is in a frame rotating at ω 0. single-spin product operator evolving into a two-spin operator, I I I. The converse is displaed in the middle illustration in which I I I. The bottom illustration in Fig. 8 shows the case in which both arrows in each of the spin states lie in the - plane. In the top two illustrations, the precession rate of a spin in the - plane depends on the up or down orientation of the other spin. In the bottom illustration the other spin is half wa between up and down, and the net result for the orientation dependence of this state is that the densit matri of the miture of the two double-arrow states does not evolve in time. This assertion is discussed in more detail in the Appendi. 9

20 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 9: COSY pulse sequence VII. VISUALIZING SPIN EVOLUTION IN A COSY PULSE SEQUENCE A. The COSY sequence As a demonstration of how double-arrow states can be used to understand the effects of pulse sequences on spin pairs, I discuss in this section the well-known correlation spectroscop, or COSY, eperiment. In a COSY eperiment molecules with two weakl interacting spins, like those discussed in Section III, are subject to two π/ pulses with a variable dela between the pulses, as illustrated in Fig. 9. This eperiment is analed using product operators in man tets see, for eample, []). ecause COSY eperiments are so widel discussed elsewhere I will not dwell on the implications of such sequences in terms of eperimental results of two-dimensional spectra; I will limit mself to a brief discussion of the time evolution of the sstem in terms of double-arrow states. 0

21 . Thermal equilibrium ensemble of spin pairs NMR eperiments begin with samples in thermal equilibrium. The approimate densit matri corresponding to an ensemble of weakl interacting spin pairs in thermal equilibrium is where, ρ eq 0 0 0, 7) γ 0 k T, 8) and the assumption has been made that. This is a diagonal densit matri, which suggests a miture of states given b the diagonal elements: spin pairs that are in the state +;+ with probabilit +)/4, in the state ; with probabilit )/4, and in the states +; and ;+ with equal probabilit /4. This interpretation of this densit matri, however, is not unique a miture of energ eigenstates is not the onl miture that will ield the equilibrium densit matri. It is especiall fruitful to rewrite the equilibrium densit matri in terms of a miture of classicall-evolving double-arrow states. The densit matri of Eq. 7) can be written as a miture of the double-arrow states illustrated in Fig. 0 plus the miture represented b the identit matri); this miture is ρ eq = 4 )E + 4 ρ +,+ + 4 ρ +, + 4 ρ +,+ + 4 ρ,+, 9) where ρ +,+ is the densit matri representation of the pure state +;+, etc. The first term in Eq. 9) is proportional to the identit matri, indicating a miture of equal populations in all four conventional basis states; such a term contributes nothing to NMR observables.) It is straightforward to demonstrate the equivalence of Eqs. 9) and 7) b writing out the densit matrices of the pure-state terms directl, but for those familiar with the product operator formalism it is perhaps easier to rewrite the equilibrium densit matri as ρ eq = 4 E + 4 I + 4 I, 30)

22 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. 0: Double-arrow states that can be used in a miture to represent thermal equilibrium. and then use Eqs. 7a) and 7b) to rewrite ρ eq in terms of the densit matrices of pure double-arrow states. The specific miture given in Eqs. 9) & 30) is not the onl wa to rewrite the equilibrium densit matri, and it is reasonable to ask wh this would be a good starting point. In the following section I anale a multiple-pulse NMR eperiment, and demonstrate that this initial state leads to a sequence of classicall evolving double-arrow states. As will be illustrated in the following section in the analsis of an eperiment with a sequence of rf pulses, this choice leads to a sequence of the special case double arrow states with classical time evolution.

23 Miture of +;+ and ;+ J coupling: πj I t = /J I I πj I I t = /J I Miture of +;+ and ; J coupling: πj πj I I I I Miture of +;+ and ; J coupling: FIG. : Representation of the COSY time evolution of the miture corresponding to the I term in the product operator epansion. C. Evolution of double-arrow states in a COSY sequence In this section I illustrate the time evolution of the miture of double-arrow states +;+ and +, in a COSY sequence; these are the pure double-arrow states associated with the product operator I. The evolution of the miture of +;+ and,+ associated with I is identical, save for a swap of indices.) For ease of visualiation I will work in a frame rotating with a frequenc ω, 0 i.e., in a frame in which there is ero offset from the precession frequenc of an isolated nucleus. Generaliation to an arbitrar offset requires additional geometric analsis, but no additional phsics. The time evolution of the double-arrow states is illustrated in Fig.. The letters correspond to various points in the pulse sequence illustrated in Fig. 9. The thermal equilibrium 3

24 miture of +;+ and +, is illustrated in Part A of the figure. The first π/) pulse rotates all arrows clockwise around the -ais, resulting in both arrows for nucleus pointing pointing in the + direction, and the arrows of nucleus pointing in the + and directions as is illustrated in Part. This result is another miture of two double-arrow states that both have simple classical-like time evolution. The thermal equilibrium densit matri could have been written in terms of a miture of +;+ and +,, but the result of the first rf pulse would have been a miture of double-arrow states with both vectors in the plane perpendicular to the field. Such states do not ehibit simple classical evolution or quantum evolution see the Appendi for more detailed analsis of the quantum evolution). In this sense, classical intuition can be used as a guide to the selection of mitures of quantum states that will be eas to anale. The results using the second version of the equilibrium densit matri won t be wrong; the just won t be as eas to anale. During the period of free evolution between the rf pulses, the dotted-line red arrow of nucleus precesses at a lab frequenc of magnitude ω 0 πj, because it represents a pure state of nucleus with a spin-up nucleus, and the solid blue arrow arrow of nucleus precesses at a lab frequenc of magnitude ω 0 + πj, because it represents a pure state of nucleus with a spin-down nucleus. In the frame rotating with frequenc ω, 0 these arrows move apart, with individual frequencies ±πj. At time time t C) the make angles ±πjt with respect to the -ais. The arrows for nucleus represent stationar states, so the do not evolve during this period. The second π/) pulse once again rotates all arrows clockwise in cones around the -ais. This brings the arrows for nucleus to the + and aes, and it rotates the arrows for nucleus into the - plane. The arrows for nucleus now form angles of ±πjt with respect to the -ais. At point D after the second rf pulse, the sstem is in a miture of double-arrow states that don t evolve in a simple wa. Note that interacting classical moments with the same orientations don t ehibit simple precession either.) At this point it is necessar to decomposethemitureintoanewmitureofstatesthatdoevolveinsimplewas, asisdiscussedin Section V. In terms of pure states, we can decompose the miture of double-arrow states labeled D in Fig. into the miture labeled D. In terms of product operators, the miture of the blue dotted-line double-arrow state and the red dotted-line double-arrow state corresponds to the product-operator I, and the miture of the solid blue double-arrow state 4

25 and the solid red double-arrow state corresponds to the product-operator I I. The relative contributions of the dotted-line states and the solid-line states in the decomposition are determined b the techniques of section V, and given b the geometr in part D of the figure: ρ D = cosπjt )I sinπjt )I I. 3) During the period of free evolution after D we can use the results discussed in Section VI. The double-arrow states corresponding to I drawn with dotted lines) rotate clockwise at Ω 0 +πj, and the double-arrows corresponding to I I do not precess. If the offset of nucleus, Ω 0, is not ero, the geometr of the arrows in illustrations like those in Fig. becomes more complicated. The arrows of nucleus are not separated smmetricall at C and D, meaning that the decomposition resulting in the miture of states represented in D will have more states in it, corresponding to additional product operator terms. This complication introduces no new phsics. VIII. CONCLUSION A complete understanding of the behavior of a microscopic sstem, like a molecule with coupled spins, requires quantum mechanics. This fact does not preclude, however, a quantum sstem from evolving in some instances in a wa that parallels that of a sstem governed b the laws of classical mechanics. In some sstems the classical quantum parallels are qualitative; in the time evolution of the quantum double-arrow states of coupled spins and mitures of these states) that I have discussed in this paper, the parallels can be quantitative and complete. complete, I mean that given an initial quantum description of an ensemble, the final quantum description of the ensemble can be determined solel using a classical analog of the sstem. The eistence of a classical analog for ensembles of weakl-coupled spins can help guide the intuition of the NMR epert sophisticated in the use of quantum mechanics. For the novice at other end of the spectrum of epertise, the classical analog provides a straightforward wa to visualie the time evolution of correlated spins that is free of the quantum machiner of product operators, unitar transformations of operators, and commutation relations. The pedagogical challenge becomes one of justifing to the novice the initial state as a representation of thermal equilibrium.) The classical evolution offers the potential to introduce the novice to multiple-pulse NMR spectroscop b focusing on 5

26 concepts, while leaving the development of quantum mechanical tools to a later time. Appendi A: Quantum mechanics of precession. Precession of single spins The quantum state θ,φ of a single spin represented b an arrow oriented in the direction given b the angles θ and φ was detailed in Eq. 3). This state is a linear combination of eigenstates of the Hamiltonian for a magnetic moment in a field in the direction. The time evolution of such eigenstates is simple: the time dependence of each eigenstate is contained in multiplicative phase factor, e ie jt/, where E j is the energ of the eigenstate. Thus, for a moment in a magnetic field, the initial state ψ0) = θ,φ evolves into the state ψt) = cos θ t/ e+iω0 + +e iφ sin θ t/ e iω0 = e iω 0t/ cos θ + +sin θ ) eiφ iω 0t) = e iω0t/ θ,φ ω 0 t cos e iω0 t/ θ e iφ ω0t) sin θ. A) This time-dependent state corresponds to an arrow-state precessing at a rate ω 0 = γ 0 at a constant inclination angle θ with respect to the direction of the field. The aimuthal angle decreases with time, corresponding to precession in a clockwise direction, as viewed from the + direction. The overall phase factor e iω0t/ is inconsequential, and disappears in the densit matri representation of the state. This phase has nothing to do with the phase angle of precession.). Precession of spin pairs a. Quantum evolution of general double-arrow spin-pair states The general form of a double-arrow state is given in Eq. 9). As in the case of a single spin, this is a linear combination of energ eigenstates, and again it is straightforward to determine the time evolution using multiplicative phase factors for each energ eigenstate 6

27 in the combination. An initial double-arrow state ψ0) = θ A,φ A ;θ,φ evolves into the state ψt) = cos θ A cos θ ) e i ω0 A ω0 +πj)t/ +;+ + e iφ cos θ A sin θ ) e i ω0 A +ω0 πj)t/ +; + e iφ A sin θ ) A e iω0 A ω0 πj)t/ ;+ + cos θ e iφ A+φ ) sin θ A sin θ ) e iω0 A +ω0 +πj)t/ ; ) +; + [ = e iω0 A +ω0 πj)t/ cos θ A cos θ + cos θ A sin θ ) e i[φ ω 0 πj)t] +; + sin θ A cos θ ) e i[φ A ωa 0 πj)t] ;+ + sin θ A sin θ ) ] e iφ A ωa 0 +φ ω 0 t) ; cos θ A cos θ e iω0 +ω0 πj)t/ e i[φ ω 0 πj)t] cos θ A sin θ. A) e i[φ A ωa 0 πj)t] sin θ A cos θ e iφ A ωa 0 t+φ ω 0 t) sin θ A sin θ b. Special Case I: Quantum evolution of uncoupled spin-pair states When the spins aren t coupled, i.e., when J = 0, the initial product state evolves to [ ψt) J=0 = e iω0 A +ω0 )t/ cos θ A cos θ ) +; + + cos θ A sin θ ) e iφ ω 0 t) +; + sin θ ) A e iφ A ωa 0 t)] ;+ + cos θ sin θ A sin θ = e iω0 A +ω0 )t/ cos θ A ) e iφ A ω 0 A t+φ ω 0 t) ; + A +e iφa ω0 t) A sin θ ) A cos θ + +e iφ ω0 t) sin θ 7 ] ), A3)

28 which is the direct product of two independentl precessing single-arrow states. This shows that the two arrows of a double-arrow quantum state precess independentl at their classical frequencies in the absence of spin-spin coupling. c. Special Case II: Coupled spin pairs one arrow along direction of field When the spins are coupled, the general result factors into independent single-arrow states if one of the single-arrow states lies along the -ais, i.e., θ A = 0, or θ A = π, or θ = 0, or θ = π. When θ A = 0, the general time-dependent state becomes ψt) θa =0 = e iω0 A +ω0 πj)t/ cos θ +;+ +sin θ ) ei[φ ω0 πj)t] +; = e iω0 A +ω0 πj)t/ + A cos θ + +e i[φa ω0 πj)t] sin θ ). A4) This is the direct product of a stationar single-arrow state oriented in the + direction, and a single-arrow state precessing with an angular frequenc ω 0 πj and again there is an inconsequential overall phase factor). For the state with θ A = π, similar analsis shows that spin precesses with a frequenc ω 0 + πj. Analogous results hold for θ = 0 and θ = π. d. Special Case III: Coupled Spins both arrows perpendicular to direction of magnetic field When coupled spins are in a state with both arrows oriented perpendicular to the direction ofthemagneticfield, i.e., θ A = θ = π/, thegeneraltime-dependentstategivenineq.a) reduces to ψt) θa =θ =π/ = [ eiω0 A +ω0 )t/ +;+ +e i[φ ω 0 πj)t] +; ] + e i[φ A ωa 0 πj)t] ;+ +e iφ A ωa 0 t+φ ω 0 t) ;. A5) This state does not factor into two single-arrow states ehibiting simple precession. For eample, the spins ma start in the factored state ψ0) = +;+, but the do not evolve into a state that can be factored, i.e., ψt) θ =π/; φ = ωt 0 8 θ =π/; φ =π/ ωt 0. A6)

29 However, a miture of two smmetricall oriented states gives a result that does ehibit simple precession. For eample, the terms in the densit matri for a miture of the pure states +; + and ; do factor. If we write time-dependent densit matri for the pure double-arrow state that starts in +;+ as ρ +,+ t), etc., we find 0 0 ie iω0 +ω0 )t [ρ +,+t)+ρ, t)] = 0 ie iω0 ω0 )t ie i ω0 +ω0 )t 0 +ie iω0 +ω0 )t 0 0 A7) Notice that the result does not contain J, the spin-coupling constant. Comparing this to the densit matri of a miture of independent spins i.e., without an coupling), we find the same result, i.e., [ ρ ) +t) ρ ) +t)+ρ ) t) ρ ) t) ] = [ρ +,+t)+ρ, t)] A8) where ρ ) +t) is the time-dependent densit matri for a single spin initiall oriented in the + direction, and precessing with angular frequenc ω 0, etc. The equivalence given in Eq. A8) justifies using the classical free evolution of the arrows in double-arrow states when both arrows are in the - plane, as long as the states are in a 50:50 miture of states with oppositel oriented arrows. This is equivalent to saing that the product operator term I I does not evolve under J coupling.) Acknowledgments I thank David Rovnak of ucknell s Chemistr Department for discussions over several ears that cataled this work, and that helped me bridge the gap between the language of chemists and phsicists. I also thank him for locating several outlets for public presentation of earl versions of this work to the NMR communit. [] Rigden JS. Quantum states and precession: The two discoveries of NMR. Rev. Mod. Phs. 986;58: [] Levitt M. Spin Dnamics, asics of Nuclear Magnetic Resonance, nd ed. West Susse, England: John Wile & Sons;

30 [3] Hanson LG. Is quantum mechanics necessar for understanding magnetic resonance? Concepts Magn. Res. Part A 008;3A: [4] Griffiths DJ. Introduction to Quantum Mechanics nd ed. Upper Saddle River, NJ: Pearson Prentice Hall, 005, Problem 5.3. [5] McIntre DH. Quantum Mechanics, A Paradigm Approach oston: Pearson, 0. [6] Sørensen OW, Eich GW, Levitt MH, odenhausen G, and Ernst RR. Product operator formalism for the description of NMR pulse eperiments. Prog. NMR Spectrosc. 983;6:63 9. [7] Packer KJ and Wright KM. The use of single-spin operator basis sets in the N.M.R. spectroscop of scalar-coupled spin sstems. Mol. Phs. 983;50: [8] van de Ven FJM and C. W. Hilbers CW. A simple formalism for the description of multiplepulse eperiments: application to a weakl coupled two-spin I = /) sstem. J. Magn. Res. 983;54:5 50. [9] P.-K. Wang P-K and Slichter CP. A pictorial operator formalism for NMR coherence phenomena. ull. Magn. Res. 986;8:3 6. [0] Donne DG and Gorenstein DG. A pictorial representation of product operator formalism: Nonclassical vector diagrams for multidimensional NMR. Concepts Magn. Res. 997;9:95. [] Freeman R. A phsical picture of multiple-quantum coherence. Concepts Magn. Res. 998;0: [] endall MR and Skinner TE. Comparison and use of vector and quantum representations of J-coupled spin evolution in an IS spin sstem during RF irradiation of one spin. J. Magn. Res. 000;43: [3] Goldenberg DP. The product operator formalism: A phsical and graphical interpretation. Concepts Magn. Res. Part A. 00;A36: [4] de la Vega-Hernánde K and Antuch M. The heteronuclear single-quantum correlation HSQC) eperiment: Vectors versus product operators. J. Chem. Educ. 05;9: [5] Slichter CP. Principles of Magnetic Resonance, 3 rd ed. New York: Springer-Verlag, 990. [6] Fenman RP, Vernon FL, and Hellwarth RW. Geometrical representaton of the Schrödinger equation for solving maser problems. J. Appl. Phs. 957;8:49 5. [7] I find the name product operator, and the labeling of the terms with the same smbols 30

31 used for angular momentum, to be somewhat misleading. Each term in the product operator epansion is simpl a densit matri for a specific ensemble miture. The terms do not correspond to dnamical operators that appear in the Hamiltonian. In this paper, product operator will alwas refer to a specific densit matri, not a dnamcial operator. 3