Product Operators 6.1 A quick review of quantum mechanics


 Arabella Miles
 2 years ago
 Views:
Transcription
1 6 Product Operators The vector model, introduced in Chapter 3, is ver useful for describing basic NMR eperiments but unfortunatel is not applicable to coupled spin sstems. When it comes to twodimensional NMR man of the eperiments are onl of interest in coupled spin sstems, so we reall must have some wa of describing the behaviour of such sstems under multiplepulse eperiments. The tools we need are provided b quantum mechanics, specificall in the form of densit matri theor which is the best wa to formulate quantum mechanics for NMR. However, we do not want to get involved in a great deal of comple quantum mechanics! Luckil, there is a wa of proceeding which we can use without a deep knowledge of quantum mechanics: this is the product operator formalism. The product operator formalism is a complete and rigorous quantum mechanical description of NMR eperiments and is well suited to calculating the outcome of modern multiplepulse eperiments. One particularl appealing feature is the fact that the operators have a clear phsical meaning and that the effects of pulses and delas can be thought of as geometrical rotations, much in the same wa as we did for the vector model in Chapter A quick review of quantum mechanics n this section we will review a few ke concepts before moving on to a description of the product operator formalism. n quantum mechanics, two mathematical objects wavefunctions and operators are of central importance. The wavefunction describes the sstem of interest (such as a spin or an electron) completel; if the wavefunction is known it is possible to calculate all the properties of the sstem. The simplest eample of this that is frequentl encountered is when considering the wavefunctions which describe electrons in atoms (atomic orbitals) or molecules (molecular orbitals). One often used interpretation of such electronic wavefunctions is to sa that the square of the wavefunction gives the probabilit of finding the electron at that point. Wavefunctions are simpl mathematical functions of position, time etc. For eample, the s electron in a hdrogen atom is described b the function ep( ar), where r is the distance from the nucleus and a is a constant. n quantum mechanics, operators represent "observable quantities" such as position, momentum and energ; each observable has an operator associated with it. Operators "operate on" functions to give new functions, hence their name operator function = (new function) An eample of an operator is ( dd); in words this operator sas "differentiate with respect to ". ts effect on the function sin is Chapter 6 "Product Operators" James Keeler, 998, 2002 &
2 d ( sin )= cos d the "new function" is cos. Operators can also be simple functions, so for eample the operator 2 just means "multipl b 2 ". A mass going round a circular path possesses angular moment, represented b a vector which points perpendicular to the plane of rotation. p 6.. Spin operators A mass going round a circular path (an orbit) possesses angular momentum; it turns out that this is a vector quantit which points in a direction perpendicular to the plane of the rotation. The ,  and components of this vector can be specified, and these are the angular momenta in the ,  and directions. n quantum mechanics, there are operators which represent these three components of the angular momentum. Nuclear spins also have angular momentum associated with them called spin angular momentum. The three components of this spin angular momentum (along, and ) are represented b the operators, and Hamiltonians The Hamiltonian, H, is the special name given to the operator for the energ of the sstem. This operator is eceptionall important as its eigenvalues and eigenfunctions are the "energ levels" of the sstem, and it is transitions between these energ levels which are detected in spectroscop. To understand the spectrum, therefore, it is necessar to have a knowledge of the energ levels and this in turn requires a knowledge of the Hamiltonian operator. n NMR, the Hamiltonian is seen as having a more subtle effect than simpl determining the energ levels. This comes about because the Hamiltonian also affects how the spin sstem evolves in time. B altering the Hamiltonian the time evolution of the spins can be manipulated and it is precisel this that lies at the heart of multiplepulse NMR. The precise mathematical form of the Hamiltonian is found b first writing down an epression for the energ of the sstem using classical mechanics and then "translating" this into quantum mechanical form according to a set of rules. n this chapter the form of the relevant Hamiltonians will simpl be stated rather than derived. n NMR the Hamiltonian changes depending on the eperimental situation. There is one Hamiltonian for the spin or spins in the presence of the applied magnetic field, but this Hamiltonian changes when a radiofrequenc pulse is applied. 6.2 Operators for one spin 6.2. Operators n quantum mechanics operators represent observable quantities, such an energ, angular momentum and magnetiation. For a single spinhalf, the   and components of the magnetiation are represented b the spin angular momentum operators, and respectivel. Thus at an time the state of the spin sstem, in quantum mechanics the densit operator, σ, can be 6 2
3 represented as a sum of different amounts of these three operators σ()= t at () + bt () + ct () The amounts of the three operators will var with time during pulses and delas. This epression of the densit operator as a combination of the spin angular momentum operators is eactl analogous to specifing the three components of a magnetiation vector. At equilibrium the densit operator is proportional to (there is onl  magnetiation present). The constant of proportionalit is usuall unimportant, so it is usual to write σ eq = 6..2 Hamiltonians for pulses and delas n order to work out how the densit operator varies with time we need to know the Hamiltonian (which is also an operator) which is acting during that time. The free precession Hamiltonian (i.e. that for a dela), H free, is H free = Ω n the vector model free precession involves a rotation at frequenc Ω about the ais; in the quantum mechanical picture the Hamiltonian involves the  angular momentum operator, there is a direct correspondence. The Hamiltonian for a pulse about the ais, H pulse, is Hpulse, = ω and for a pulse about the ais it is Hpulse, = ω Again there is a clear connection to the vector model where pulses result in rotations about the  or aes Equation of motion The densit operator at time t, σ(t), is computed from that at time 0, σ(0), using the following relationship σ()= t ep( iht) σ( 0) ep( iht) where H is the relevant hamiltonian. f H and σ are epressed in terms of the angular momentum operators if turns out that this equation can be solved easil with the aid of a few rules. Suppose that an pulse, of duration t p, is applied to equilibrium magnetiation. n this situation H = ω and σ(0) = ; the equation to be solved is σ( tp)= ep( iωtp) ep( iωtp ) Such equations involving angular momentum operators are common in quantum mechanics and the solution to them are alread all know. The identit required here to solve this equation is ( ) ( ) ep iβ ep iβ cosβ sin β [6.] This is interpreted as a rotation of b an angle β about the ais. B 6 3
4 putting β= ω t p this identit can be used to solve Eqn. [6.] σ( tp)= cosωtp sinωtp The result is eactl as epected from the vector model: a pulse about the  ais rotates magnetiation towards the ais, with a sinusoidal dependence on the flip angle, β Standard rotations Given that there are onl three operators, there are a limited number of identities of the tpe of Eqn. [6.]. The all have the same form ( ) ( ) ep iθa {old operator} ep iθa cosθ {old operator} + sinθ {new operator} where {old operator}, {new operator} and a are determined from the three possible angular momentum operators according to the following diagrams; the label in the centre indicates which ais the rotation is about 6 4 Angle of rotation = Ωt for offsets and ω t p for pulses First eample: find the result of rotating the operator b θ about the ais, that is ( ) ( ) ep iθ ep iθ For rotations about the middle diagram is required. The diagram shows that (the "old operator") is rotated to (the "new operator"). The required identit is therefore ( ) ( ) + ep iθ ep iθ cosθ sinθ Second eample: find the result of ( ){ } ( ) ep iθ ep iθ This is a rotation about, so diagram is required. The diagram shows that (the "old operator") is rotated to (the "new operator"). The required identit is therefore ep iθ ep iθ cosθ sin θ ( ) { } ( ) { }+ { } cosθ sinθ Finall, note that a rotation of an operator about its own ais has no effect e.g. a rotation of about leaves unaltered Shorthand notation To save writing, the arrow notation is often used. n this, the term Ht is written over an arrow which connects the old and new densit operators. So, for eample, the following
5 is written σ( tp)= ep iωtp σ 0 ep iωtp ( ) ( ) ( ) ωt p σ( 0) σ t ( p) For the case where σ(0) = p cos ω t sinω t ω t 6..6 Eample calculation: spin echo p p a b e f 90 delaτ 80 delaτ acquire ( ) ( ) At a the densit operator is. The transformation from a to b is free precession, for which the Hamiltonian is Ω ; the dela τ therefore corresponds to a rotation about the ais at frequenc Ω. n the shorthand notation this is Ωτ σ b ( ) To solve this diagram above is needed with the angle = Ωτ; the "new operator" is Ωτ cosωτ + sin Ωτ n words this sas that the magnetiation precesses from towards +. The pulse about has the Hamiltonian ω ; the pulse therefore corresponds to a rotation about for a time t p such that the angle, ω t p, is π radians. n the shorthand notation ω cosωτ + sin Ωτ p t σ() e [6.2] Each term on the left is dealt with separatel. The first term is a rotation of about ; the relevant diagram is thus ω t p cosωτ cosωτcosω t cosωτsinω t p p However, the flip angle of the pulse, ω t p, is π so the second term on the right is ero and the first term just changes sign (cos π = ); overall the result is π cosωτ cosωτ The second term on the left of Eqn. [6.2] is eas to handle as it is unaffected b a rotation about. Overall, the effect of the 80 pulse is then π cosωτ + sin Ωτ cosωτ + sin Ωτ [6.3] As was shown using the vector model, the component just changes sign. The net stage is the evolution of the offset for time τ. Again, each term on the right of Eqn. [6.3] is considered separatel Ωτ cosωτ cosωτcosωτ sin ΩτcosΩτ Ωτ sin Ωτ cosωτsin Ωτ + sin Ωτsin Ωτ Collecting together the terms in and the final result is cosωτcosωτ + sin Ωτsin Ωτ cosωτsin Ωτ sin ΩτcosΩτ ( ) + ( ) 6 5
6 The bracket multipling is ero and the bracket multipling is = because of the identit cos 2 θ + sin 2 θ =. Thus the overall result of the spin echo sequence can be summarised 90 ( ) 80 τ ( ) τ n words, the outcome is independent of the offset, Ω, and the dela τ, even though there is evolution during the delas. The offset is said to be refocused b the spin echo. This is eactl the result we found in section 3.8. n general the sequence τ 80 () τ [6.4] refocuses an evolution due to offsets; this is a ver useful feature which is much used in multiplepulse NMR eperiments. One further point is that as far as the offset is concerned the spin echo sequence of Eqn. [6.4] is just equivalent to 80 (). J 2 J 2 Ω Ω 2 The spectrum from two coupled spins, with offsets Ω and Ω 2 (rad s ) and mutual coupling J 2 (H). 6.2 Operators for two spins The product operator approach comes into its own when coupled spin sstems are considered; such sstems cannot be treated b the vector model. However, product operators provide a clean and simple description of the important phenomena of coherence transfer and multiple quantum coherence. 6.. Product operators for two spins For a single spin the three operators needed for a complete description are, and. For two spins, three such operators are needed for each spin; an additional subscript, or 2, indicates which spin the refer to. spin : spin 2 : represents magnetiation of spin, and 2 likewise for spin 2. represents magnetiation on spin. As spin and 2 are coupled, the spectrum consists of two doublets and the operator can be further identified with the two lines of the spin doublet. n the language of product operators is said to represent inphase magnetiation of spin ; the description inphase means that the two lines of the spin doublet have the same sign and lineshape. Following on in the same wa 2 represents inphase magnetiation on spin 2. and 2 also represent inphase magnetiation on spins and 2, respectivel, but this magnetiation is aligned along and so will give rise to a different lineshape. Arbitraril, an absorption mode lineshape will be assigned to magnetiation aligned along and a dispersion mode lineshape to magnetiation along. 6 6
7 absorption dispersion Ω Ω 2 2 Ω 2 There are four additional operators which represent antiphase magnetiation: 2 2, 2 2, 2 2, 2 2 (the factors of 2 are needed for normaliation purposes). The operator 2 2 is described as magnetiation on spin which is antiphase with respect to the coupling to spin 2. Ω Ω Ω 2 The absorption and dispersion lineshapes. The absorption lineshape is a maimum on resonance, whereas the dispersion goes through ero at this point. The "cartoon" forms of the lineshapes are shown in the lower part of the diagram. Ω 2 Note that the two lines of the spin multiplet are associated with different spin states of spin2, and that in an antiphase multiplet these two lines have different signs. Antiphase terms are thus sensitive to the spin states of the coupled spins. There are four remaining product operators which contain two transverse (i.e.  or operators) terms and correspond to multiplequantum coherences; the are not observable multiple quantum : Ω Finall there is the term 2 2 which is also not observable and corresponds to a particular kind of nonequilibrium population distribution. α Ω β spin state of spin 2 The two lines of the spin doublet can be associated with different spin states of spin Evolution under offsets and pulses The operators for two spins evolve under offsets and pulses in the same wa as do those for a single spin. The rotations have to be applied separatel to each spin and it must be remembered that rotations of spin do not affect spin 2, and vice versa. For eample, consider evolving under the offset of spin and spin 2. The relevant Hamiltonian is H = Ω + Ω free 22 where Ω and Ω 2 are the offsets of spin and spin 2 respectivel. Evolution under this Hamiltonian can be considered b appling the two terms sequentiall (the order is immaterial) Hfreet Ωt+ Ω2t2 Ωt Ω2t2 The first "arrow" is a rotation about Ωt Ω2t2 cosω t + sin Ω t 6 7
8 The second arrow leaves the intermediate state unaltered as spin2 operators have not effect on spin operators. Overall, therefore Ωt+ Ω2t2 cosω t + sin Ω t A second eample is the term 2 2 evolving under a 90 pulse about the ais applied to both spins. The relevant Hamiltonian is H = ω + ω 2 The evolution can be separated into two successive rotations ωt ωt2 2 2 The first arrow affects onl the spin operators; a 90 rotation of about gives (remembering that ω t = π/2 for a 90 pulse) ω t cosω t2 sinω t π 2 ω t 2 The second arrow onl affects the spin 2 operators; a 90 rotation of about takes it to π The overall result is that antiphase magnetiation of spin has been transferred into antiphase magnetiation of spin 2. Such a process is called coherence transfer and is eceptionall important in multiplepulse NMR Evolution under coupling The new feature which arises when considering two spins is the effect of coupling between them. The Hamiltonian representing this coupling is itself a product of two operators: H = 2 J πj2 2 where J 2 is the coupling in H. Evolution under coupling causes the interconversion of inphase and antiphase magnetiation according to the following diagrams V π 2 V π 2 angle = πjt For eample, inphase magnetiation along becomes antiphase along according to the diagram V 2πJ t 2 2 cosπj t + sinπj t note that the angle is πj 2 t i.e. half the angle for the other rotations,. Antiphase magnetiation along becomes inphase magnetiation along ; using diagram V: 2πJ2t 2 2 cosπj t2 + sinπj t
9 The diagrams appl equall well to spin2; for eample 2πJ2t 2 2 cosπj t2 + sinπj t Complete interconversion of inphase and antiphase magnetiation requires a dela such that πj2t = π 2 i.e. a dela of /(2J 2 ). A dela of /J 2 causes inphase magnetiation to change its sign: 2πJ2t 2 t= 2J2 2πJ2t 2 t= J Spin echoes t was shown in section that the offset is refocused in a spin echo. n this section it will be shown that the evolution of the scalar coupling is not necessaril refocused Spin echoes in homonuclear spin sstem n this kind of spin echo the 80 pulse affects both spins i.e. it is a nonselective pulse: τ 80 (, to spin and spin 2) τ At the start of the sequence it will be assumed that onl inphase  magnetiation on spin is present:. n fact the starting state is not important to the overall effect of the spin echo, so this choice is arbitrar. t was shown in section that the spin echo applied to one spin refocuses the offset; this conclusion is not altered b the presence of a coupling so the offset will be ignored in the present calculation. This greatl simplifies things. For the first dela τ onl the effect of evolution under coupling need be considered therefore: 2πJ2τ 2 cosπj τ + sinπj τ The 80 pulse affects both spins, and this can be calculated b appling the 80 rotation to each in succession π π2 cosπj τ + sinπj τ where it has alread been written in that ω t p = π, for a 80 pulse. The 80 rotation about for spin has no effect on the operator and 2, and it simpl reverses the sign of the operator π π2 cosπj τ + sinπj τ2 cosπj τ sinπj τ The 80 rotation about for spin 2 has no effect on the operators and, but simpl reverses the sign of the operator 2. The final result is thus π cosπj τ + sinπj τ2 cosπj τ sinπj τ π2 cosπj τ + sinπj τ Nothing has happened; the 80 pulse has left the operators unaffected! So, for the purposes of the calculation it is permissible to ignore the 80 pulse and simpl allow the coupling to evolve for 2τ. The final result can therefore just be written down: 2 6 9
10 τ 80 ( ) τ cos2πj τ + sin 2πJ τ From this it is eas to see that complete conversion to antiphase magnetiation requires 2πJ 2 τ = π/2 i.e. τ = /(4 J 2 ). The calculation is not quite as simple if the initial state is chosen as, but the final result is just the same the coupling evolves for 2τ: τ 80 ( ) τ cos 2πJ τ + sin 2πJ τ 2 n fact, the general result is that the sequence τ 80 (, to spin and spin 2) τ is equivalent to the sequence 2τ 80 (, to spin and spin 2) in which the offset is ignored and coupling is allowed to act for time 2τ nterconverting inphase and antiphase states So far, spin echoes have been demonstrated as being useful for generating antiphase terms, independent of offsets. For eample, the sequence 90 () /(4J 2 ) 80 () /(4J 2 ) generates pure antiphase magnetiation. Equall useful is the sequence /(4J 2 ) 80 () /(4J 2 ) which will convert pure antiphase magnetiation, such as 2 2 into inphase magnetiation,. a spin spin 2 b spin spin 2 c spin spin 2 τ τ τ Three different spin echo sequences that can be applied to heteronuclear spin sstems. The open rectangles represent 80 pulses. τ τ τ 6..3 Spin echoes in heteronuclear spin sstems f spin and spin 2 are different nuclear species, such as 3 C and H, it is possible to choose to appl the 80 pulse to either or both spins; the outcome of the sequence depends on the pattern of 80 pulses. Sequence a has alread been analsed: the result is that the offset is refocused but that the coupling evolves for time 2τ. Sequence b still refocuses the offset of spin, but it turns out that the coupling is also refocused. Sequence c refocuses the coupling but leaves the evolution of the offset unaffected. Sequence b t will be assumed that the offset is refocused, and attention will therefore be restricted to the effect of the coupling 2πJ2τ 2 cosπj τ + sinπj τ 2 The 80 () pulse is onl applied to spin π cosπj τ + sinπj τ 2 cosπj τ sinπj τ 2 [6.5] The two terms on the right each evolve under the coupling during the second dela: 6 0
11 cosπj 2πJ2τ 2 τ cosπj τcosπj τ + sinπj τcosπj τ πJ2τ 2 sinπj2τ 22 cosπj2τsinπj2τ 22 + sinπj2τ sinπj2τ Collecting the terms together and noting that cos 2 θ + sin 2 θ = the final result is just. n words, the effect of the coupling has been refocused. Sequence c As there is no 80 pulse applied to spin, the offset of spin is not refocused, but continues to evolve for time 2τ. The evolution of the coupling is eas to calculate: 2πJ2τ 2 cosπj τ + sinπj τ 2 This time the 80 () pulse is applied to spin π cosπj τ + sinπj τ 2 2 cosπj τ sinπj τ The results is eactl as for sequence b (Eqn. [6.5]), so the final result is the same i.e. the coupling is refocused. Summar n heteronuclear sstems it is possible to choose whether or not to allow the offset and the coupling to evolve; this gives great freedom in generating and manipulating antiphase states which pla a ke role in multiple pulse NMR eperiments. 6.4 Multiple quantum terms 6.4. Coherence order n NMR the directl observable quantit is the transverse magnetiation, which in product operators is represented b terms such as and 2 2. Such terms are eamples of single quantum coherences, or more generall coherences with order, p, = ±. Other product operators can also be classified according to coherence order e.g. 2 2 has p = 0 and 2 2 has both p = 0 (eroquantum coherence) and ±2 (double quantum coherence). Onl single quantum coherences are observable. n heteronuclear sstems it is sometimes useful to classif operators according to their coherence orders with respect to each spin. So, for eample, 2 2 has p = 0 for spin and p = ± for spin Raising and lowering operators The classification of operators according to coherence order is best carried out be reepressing the Cartesian operators and in terms of the raising and lowering operators, + and, respectivel. These are defined as follows = + i = i [6.6] + where i is the square root of. + has coherence order + and has coherence order ; coherence order is a signed quantit. 6
12 Using the definitions of Eqn. [6.6] and can be epressed in terms of the raising and lowering operators ( ) = ( ) = i + [6.7] from which it is seen that and are both mitures of coherences with p = + and. The operator product 2 2 can be epressed in terms of the raising and lowering operators in the following wa (note that separate operators are used for each spin: ± and 2± ) 2 = 2 2 ( 2 + ) ( ) [6.8] = 2 ( )+ 2 ( ) The first term on the right of Eqn. [6.8] has p = (++) = 2 and the second term has p = ( ) = 2; both are double quantum coherences. The third and fourth terms both have p = (+ ) = 0 and are ero quantum coherences. The value of p can be found simpl b noting the number of raising and lowering operators in the product. The pure double quantum part of 2 2 is, from Eqn. [6.8], double quantum part[ 2 2 ]= 2 ( ) [6.9] The raising and lowering operators on the right of Eqn. [6.9] can be reepressed in terms of the Cartesian operators: 2 ( )= 2 ( + i) ( 2 + i2)+ ( i) ( 2 i2) [ ] [ ] = So, the pure double quantum part of 2 2 is 2 ( ); b a similar method the pure ero quantum part can be shown to be 2 ( 22 22). Some further useful relationships are given in section 6.9 J 3 J 2 Ω α α β β spin 2 α β α β spin 3 The doublet of doublets from spin coupled to two other spins. The spin states of the coupled spins are also indicated. 6.5 Three spins The product operator formalism can be etended to three or more spins. No reall new features arise, but some of the ke ideas will be highlighted in this section. The description will assume that spin is coupled to spins 2 and 3 with coupling constants J 2 and J 3 ; in the diagrams it will be assumed that J 2 > J Tpes of operators represents inphase magnetiation on spin ; 2 2 represents magnetiation antiphase with respect to the coupling to spin 2 and 2 3 represents magnetiation antiphase with respect to the coupling to spin represents magnetiation which is doubl antiphase with respect to the couplings to both spins 2 and
13 As in the case of two spins, the presence of more than one transverse operator in the product represents multiple quantum coherence. For eample, 2 2 is a miture of double and eroquantum coherence between spins and 2. The product is the same miture, but antiphase with respect to the coupling to spin 3. Products such as contain, amongst other things, triplequantum coherences Evolution Evolution under offsets and pulses is simpl a matter of appling sequentiall the relevant rotations for each spin, remembering that rotations of spin do not affect operators of spins 2 and 3. For eample, the term 2 2 evolves under the offset in the following wa: Ωt Ω2t2 Ω3t3 2 cosω t 2 + sin Ω t The first arrow, representing evolution under the offset of spin, affects onl the spin operator. The second arrow has no effect as the spin 2 operator 2 and this is unaffected b a rotation. The third arrow also has no effect as there are no spin 3 operators present. The evolution under coupling follows the same rules as for a twospin sstem. For eample, evolution of under the influence of the coupling to spin 3 generates 2 3 2πJ t 3 3 cosπj t + sinπj t Further evolution of the term 2 3 under the influence of the coupling to spin 2 generates a double antiphase term 2πJ2t 2 2 cosπj t 2 sinπj t n this evolution the spin 3 operator if unaffected as the coupling does not involve this spin. The connection with the evolution of under a coupling can be made more eplicit b writing 2 3 as a "constant" γ 2πJ t γ 2 2 cosπj t γ sinπj t 2 γ which compares directl to πJ t 2 2 cosπj t sinπj t Representations of different tpes of operators. 6.6 Alternative notation n this chapter different spins have been designated with a subscript, 2, 3... Another common notation is to distinguish the spins b using a different letter to represent their operators; commonl and S are used for two of the smbols 22 2S Note that the order in which the operators are written is not important, although it is often convenient (and tid) alwas to write them in the same sequence. n heteronuclear eperiments a notation is sometimes used where the letter represents the nucleus. So, for eample, operators referring to protons are given the letter H, carbon3 atoms the letter C and nitrogen5 atoms 6 3
14 the letter N; carbonl carbons are sometimes denoted C'. For eample, 4C H N denotes magnetiation on carbon3 which is antiphase with respect to coupling to both proton and nitrogen Conclusion The product operator method as described here onl applies to spinhalf nuclei. t can be etended to higher spins, but significant etra compleit is introduced; details can be found in the article b Sørensen et al. (Prog. NMR Spectrosc. 6, 63 (983)). The main difficult with the product operator method is that the more pulses and delas that are introduced the greater becomes the number of operators and the more comple the trigonometrical epressions multipling them. f pulses are either 90 or 80 then there is some simplification as such pulses do not increase the number of terms. As will be seen in chapter 7, it is important to tr to simplif the calculation as much as possible, for eample b recogniing when offsets or couplings are refocused b spin echoes. A number of computer programs are available for machine computation using product operators within programs such as Mathematica. These can be ver labour saving. 6.8 Multiple quantum coherence 6.8. Multiplequantum terms n the product operator representation of multiple quantum coherences it is usual to distinguish between active and passive spins. Active spins contribute transverse operators, such as, and +, to the product; passive spins contribute onl operators,. n a sense the spins contributing transverse operators are "involved" in the coherence, while those contributing operators are simpl spectators. For double and eroquantum coherence in which spins i and j are active it is convenient to define the following set of operators which represent pure multiple quantum states of given order. The operators can be epressed in terms of the Cartesian or raising and lowering operators. double quantum, p =± 2 ( ij) DQ 2 ( 2ij 2ij ) 2 ( i+ j+ + i j ) ( ij) DQ 2 ( 2ij + 2ij ) 2i ( i+ j+ i j ) ero quantum, p = 0 ( ij) ZQ 2 ( 2ij + 2ij ) 2 ( i+ j + i j+ ) ( ij) ZQ 2 ( 2ij 2ij ) 2 i ( i+ j i j+ ) Evolution of multiple quantum terms Evolution under offsets The double and eroquantum operators evolve under offsets in a wa 6 4
15 which is entirel analogous to the evolution of and under free precession ecept that the frequencies of evolution are (Ω i + Ω j ) and (Ω i Ω j ) respectivel: ( ij) Ωiti+ Ωjt j ( ij) ( ij) DQ cos Ω + Ω tdq sin Ω Ω tdq i i j j ij DQ cos Ω + Ω tdq sin Ω Ω tdq ZQ ( ) + ( + ) i j i j ( ij) Ωt + Ω t ( ) ( ij) ( i j) ( i + j) ( ij) Ωiti+ Ωjt j ( ij) ( ij) cos( Ωi Ωj) tzq + sin( Ωi Ωj) tzq ( ij) Ωiti+ Ωjt j ( ij) ( ij) cos Ωi Ωj t sin Ωi Ωj t ( ) ( ) ZQ ZQ ZQ Evolution under couplings Multiple quantum coherence between spins i and j does not evolve under the influence of the coupling between the two active spins, i and j. Double and eroquantum operators evolve under passive couplings in a wa which is entirel analogous to the evolution of and ; the resulting multiple quantum terms can be described as being antiphase with respect to the effective couplings: ( ij) ( ij) ( ij) DQ cosπj t DQ + cosπj t 2 DQ ij ij DQ cosπj t DQ sinπj t 2 DQ ij ij ZQ cosπj tzq + sinπj t 2 ZQ ZQ DQ,eff DQ,eff k ( ) ( ) ( ij) DQ,eff DQ,eff k ( ) ( ) ( ij) ZQ,eff ZQ,eff k ( ij) ( ij) ( ij) cosπj ZQ,efftZQ sinπjzq,efft 2kZQ J DQ,eff is the sum of the couplings between spin i and all other spins plus the sum of the couplings between spin j and all other spins. J ZQ,eff is the sum of the couplings between spin i and all other spins minus the sum of the couplings between spin j and all other spins. For eample in a threespin sstem the eroquantum coherence between spins and 2, antiphase with respect to spin 3, evolves according to ( 2) ( 2) ( 2) 23ZQ cosπjzq,efft 23ZQ sinπjzq,efft ZQ where J = J J ZQ,eff 2 23 Further details of multiplequantum evolution can be found in section 5.3 of Ernst, Bodenhausen and Wokaun Principles of NMR in One and Two Dimensions (Oford Universit Press, 987). 6 5
Pulsed Fourier Transform NMR The rotating frame of reference. The NMR Experiment. The Rotating Frame of Reference.
Pulsed Fourier Transform NR The rotating frame of reference The NR Eperiment. The Rotating Frame of Reference. When we perform a NR eperiment we disturb the equilibrium state of the sstem and then monitor
More informationThe basic 2D spectrum would involve repeating a multiple pulse 1D sequence with a systematic variation of the delay time t D
2D NMR spectroscop So far we have been dealing with multiple pulses but a single dimension  that is, 1D spectra. We have seen, however, that a multiple pulse sequence can give different spectra which
More informationSpinlattice and spinspin relaxation
Spinlattice and spinspin relaation Sequence of events in the NMR eperiment: (i) application of a 90 pulse alters the population ratios, and creates transverse magnetic field components (M () ); (ii)
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More information7 Twodimensional NMR
7 Twodimensional NMR 7. Introduction The basic ideas of twodimensional NMR will be introduced by reference to the appearance of a COSY spectrum; later in this chapter the product operator formalism will
More information3 The vector model. Chapter 3 The vector model c James Keeler, 2002 & 2004. 3.1 Bulk magnetization
3 The vector model For most kinds of spectroscop it is sufficient to think about energ levels and selection rules; this is not true for NMR. For eample, using this energ level approach we cannot even describe
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationSECTION 74 Algebraic Vectors
74 lgebraic Vectors 531 SECTIN 74 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More information1 Introduction to quantum mechanics
ntroduction to quantum mechanics Quantum mechanics is the basic tool needed to describe, understand and devise NMR experiments. Fortunately for NMR spectroscopists, the quantum mechanics of nuclear spins
More information2, 8, 20, 28, 50, 82, 126.
Chapter 5 Nuclear Shell Model 5.1 Magic Numbers The binding energies predicted by the Liquid Drop Model underestimate the actual binding energies of magic nuclei for which either the number of neutrons
More informationAddition and Subtraction of Vectors
ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b
More informationChapter 3 Vectors 3.1 Vector Analysis Introduction to Vectors Properties of Vectors Cartesian Coordinate System...
Chapter 3 Vectors 3.1 Vector Analsis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Cartesian Coordinate Sstem... 5 3.2.1 Cartesian Coordinates... 6 3.3 Application of Vectors...
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationSTRESSSTRAIN RELATIONS
STRSSSTRAIN RLATIONS Strain Strain is related to change in dimensions and shape of a material. The most elementar definition of strain is when the deformation is along one ais: change in length strain
More informationA Pictorial Representation of Product Operator Formalism: NonClassical Vector Diagrams for Multidimensional NMR
Chapter 8 A Pictorial Representation of Product Operator Formalism: NonClassical Vector Diagrams for Multidimensional NMR DAVID G. DONNE 1 AND DAVID G. GORENSTEIN Department of Molecular Biology The Scripps
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationSection 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions
Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,
More informationSTRESS TRANSFORMATION AND MOHR S CIRCLE
Chapter 5 STRESS TRANSFORMATION AND MOHR S CIRCLE 5.1 Stress Transformations and Mohr s Circle We have now shown that, in the absence of bod moments, there are si components of the stress tensor at a material
More informationIntroduction to polarization of light
Chapter 2 Introduction to polarization of light This Chapter treats the polarization of electromagnetic waves. In Section 2.1 the concept of light polarization is discussed and its Jones formalism is presented.
More informationModule in Radio Propagation
1 This chapter is a brief overview of the subject of radio waves: what the are, how the can be characterised, and how their properties are related to those of the media in which the travel. It discusses
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationQ (x 1, y 1 ) m = y 1 y 0
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
More informationNMR for Physical and Biological Scientists Thomas C. Pochapsky and Susan Sondej Pochapsky Table of Contents
Preface Symbols and fundamental constants 1. What is spectroscopy? A semiclassical description of spectroscopy Damped harmonics Quantum oscillators The spectroscopic experiment Ensembles and coherence
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationASEN Structures. Stress in 3D. ASEN 3112 Lecture 1 Slide 1
ASEN 3112  Structures Stress in 3D ASEN 3112 Lecture 1 Slide 1 ASEN 3112  Structures Mechanical Stress in 3D: Concept Mechanical stress measures intensit level of internal forces in a material (solid
More informationMATH 118, LECTURES 14 & 15: POLAR AREAS
MATH 118, LECTURES 1 & 15: POLAR AREAS 1 Polar Areas We recall from Cartesian coordinates that we could calculate the area under the curve b taking Riemann sums. We divided the region into subregions,
More informationw = COI EYE view direction vector u = w ( 010,, ) cross product with yaxis v = w u up vector
. w COI EYE view direction vector u w ( 00,, ) cross product with ais v w u up vector (EQ ) Computer Animation: Algorithms and Techniques 29 up vector view vector observer center of interest 30 Computer
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationModelling musical chords using sine waves
Modelling musical chords using sine waves Introduction From the stimulus word Harmon, I chose to look at the transmission of sound waves in music. As a keen musician mself, I was curious to understand
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More information4.1 PiecewiseDefined Functions
Section 4.1 PiecewiseDefined Functions 335 4.1 PiecewiseDefined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept
More informationClassical Angular Momentum. The Physics of Rotational Motion.
3. Angular Momentum States. We now employ the vector model to enumerate the possible number of spin angular momentum states for several commonly encountered situations in photochemistry. We shall give
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information2 NMR and energy levels
NMR and energy levels The picture that we use to understand most kinds of spectroscopy is that molecules have a set of energy levels and that the lines we see in spectra are due to transitions between
More informationSTIFFNESS OF THE HUMAN ARM
SIFFNESS OF HE HUMAN ARM his web resource combines the passive properties of muscles with the neural feedback sstem of the shortloop (spinal) and longloop (transcortical) reflees and eamine how the whole
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationPulsed Nuclear Magnetic Resonance Experiment
Pulsed Nuclear agnetic Resonance Eperiment in LI Dept. of Phsics & Astronom, Univ. of Pittsburgh, Pittsburgh, PA 56, USA arch 8 th, Introduction. Simple Resonance heor agnetic resonance is a phenomenon
More informationNuclear Magnetic Resonance Lisa M. Larrimore
Lisa M. The fundamentals of nuclear magnetic resonance (NMR) spectroscopy were explored on samples containing large numbers of protons. Mineral oil and dilluted solutions of CuSO 4 were placed in a permanent
More informationEQUILIBRIUM STRESS SYSTEMS
EQUILIBRIUM STRESS SYSTEMS Definition of stress The general definition of stress is: Stress = Force Area where the area is the crosssectional area on which the force is acting. Consider the rectangular
More informationNuclear Magnetic Resonance
Nuclear Magnetic Resonance Introduction Atomic magnetism Nuclear magnetic resonance refers to the behaviour of atomic nuclei in the presence of a magnetic field. The first principle required to understand
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationPlane Stress Transformations
6 Plane Stress Transformations ASEN 311  Structures ASEN 311 Lecture 6 Slide 1 Plane Stress State ASEN 311  Structures Recall that in a bod in plane stress, the general 3D stress state with 9 components
More informationCOMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN
COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject
More informationthat satisfies (2). Then (3) ax 0 + by 0 + cz 0 = d.
Planes.nb 1 Plotting Planes in Mathematica Copright 199, 1997, 1 b James F. Hurle, Universit of Connecticut, Department of Mathematics, Unit 39, Storrs CT 66939. All rights reserved. This notebook discusses
More informationNMR Spectroscopy. Applications. Drug design MRI. Food quality. Structural biology. Metabonomics
Applications Drug design MRI Food qualit Metabonomics Structural biolog Basic Principles N.M.R. = Nuclear Magnetic Resonance Spectroscopic technique, thus relies on the interaction between material and
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More informationNMR SPECTROSCOPY. Basic Principles, Concepts, and Applications in Chemistry. Harald Günther University of Siegen, Siegen, Germany.
NMR SPECTROSCOPY Basic Principles, Concepts, and Applications in Chemistry Harald Günther University of Siegen, Siegen, Germany Second Edition Translated by Harald Günther JOHN WILEY & SONS Chichester
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationI think that starting
. Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle  in particular the basic equation representing
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationLineTorus Intersection for Ray Tracing: Alternative Formulations
LineTorus Intersection for Ra Tracing: Alternative Formulations VACLAV SKALA Department of Computer Science Engineering Universit of West Bohemia Univerzitni 8, CZ 30614 Plzen CZECH REPUBLIC http://www.vaclavskala.eu
More informationSome linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2015
Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 25 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,
More informationEXPANDING THE CALCULUS HORIZON. Hurricane Modeling
EXPANDING THE CALCULUS HORIZON Hurricane Modeling Each ear population centers throughout the world are ravaged b hurricanes, and it is the mission of the National Hurricane Center to minimize the damage
More informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationLogic Design 2013/9/5. Introduction. Logic circuits operate on digital signals
Introduction Logic Design Chapter 2: Introduction to Logic Circuits Logic circuits operate on digital signals Unlike continuous analog signals that have an infinite number of possible values, digital signals
More informationLecture 1: STRESS IN 3D TABLE OF CONTENTS. Page
1 Stress in 3D 1 1 Lecture 1: STRESS IN 3D TABLE OF CONTENTS Page 1.1 Introduction..................... 1 3 1.2 Mechanical Stress: Concept............... 1 3 1.3 Mechanical Stress: Definition...............
More information2D Geometrical Transformations. Foley & Van Dam, Chapter 5
2D Geometrical Transformations Fole & Van Dam, Chapter 5 2D Geometrical Transformations Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2D Geometrical Transformations
More informationStructure Determination by NMR
Structure Determination by NMR * Introduction to NMR * 2D NMR, resonance assignments J Correlated Based Experiments * COSY  Correlated Spectroscopy * NOESY  Nuclear Overhauser Effect Spectroscopy * HETCOR
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationPrecession of spin and Precession of a top
6. Classical Precession of the Angular Momentum Vector A classical bar magnet (Figure 11) may lie motionless at a certain orientation in a magnetic field. However, if the bar magnet possesses angular momentum,
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationVector Calculus: a quick review
Appendi A Vector Calculus: a quick review Selected Reading H.M. Sche,. Div, Grad, Curl and all that: An informal Tet on Vector Calculus, W.W. Norton and Co., (1973). (Good phsical introduction to the subject)
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationBending of Beams with Unsymmetrical Sections
Bending of Beams with Unsmmetrical Sections Assume that CZ is a neutral ais. C = centroid of section Hence, if > 0, da has negative stress. From the diagram below, we have: δ = α and s = αρ and δ ε = =
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationCALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES
6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More information2.4 Orthogonal Coordinate Systems
8/25/2005 section 2_4Orthogonal_Coordinate_Sstems_empt.doc 1/6 2.4 Orthogonal Coordinate Sstems Reading Assignment: pp.1633 We live in a 3dimensional world! Meaning: 1) 2) Q: What 3 scalar values and
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationMathematical goals. Starting points. Materials required. Time needed
Level A7 of challenge: C A7 Interpreting functions, graphs and tables tables Mathematical goals Starting points Materials required Time needed To enable learners to understand: the relationship between
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationParticle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims
Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;
More informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More informationProton Nuclear Magnetic Resonance Spectroscopy
Proton Nuclear Magnetic Resonance Spectroscopy Introduction: The NMR Spectrum serves as a great resource in determining the structure of an organic compound by revealing the hydrogen and carbon skeleton.
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationBasic Principles of Magnetic Resonance
Basic Principles of Magnetic Resonance Contents: Jorge Jovicich jovicich@mit.edu I) Historical Background II) An MR experiment  Overview  Can we scan the subject?  The subject goes into the magnet 
More informationSolutions manual for Understanding NMR spectroscopy second edition
Solutions manual for Understanding NMR spectroscopy second edition James Keeler and Andrew J. Pell University of Cambridge, Department of Chemistry (a) (b) Ω Ω 2 Ω 2 Ω ω ω 2 Version 2.0 James Keeler and
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infrared spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More information3D Stress Components. From equilibrium principles: τ xy = τ yx, τ xz = τ zx, τ zy = τ yz. Normal Stresses. Shear Stresses
3D Stress Components From equilibrium principles:, z z, z z The most general state of stress at a point ma be represented b 6 components Normal Stresses Shear Stresses Normal stress () : the subscript
More information3 Unit Circle Trigonometry
0606_CH0_78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More information