State of a system at time t Density Operator Weve seen this before as a projection operator Can find density matrix in terms of the basis set Matrix elements of density matrix Contains time dependent ID: 1030059
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1. Chapter 14
2. Density MatrixState of a system at time t:Density OperatorWe’ve seen this before, as a “projection operator”Can find density matrix in terms of the basis set Matrix elements of density matrix:Contains time dependentphase factors.Copyright – Michael D. Fayer, 2018
3. Two state system:Calculate matrix elements of 2×2 density matrix:Time dependent phasefactors cancel. Alwayshave ket with its complexconjugate bra.Copyright – Michael D. Fayer, 2018
4. In general:ij density matrix elementCopyright – Michael D. Fayer, 2018
5. 2×2 Density Matrix:Diagonal density matrix elements probs. of finding system in various statesOff Diagonal Elements “coherences”trace = 1 for anydimension(trace – sum of diagonal matrix elements)AndCopyright – Michael D. Fayer, 2018
6. Time dependence of product ruleUsing Schrödinger Eq. for time derivatives ofCopyright – Michael D. Fayer, 2018
7. Substituting:density operatorTherefore:The fundamental equation of the density matrix representation.Copyright – Michael D. Fayer, 2018
8. Density Matrix Equations of Motionby product ruleFor 2×2 case, the equation of motion is:time derivative of density matrix elementsCopyright – Michael D. Fayer, 2018
9. Copyright – Michael D. Fayer, 2018Equations of Motion – from multiplying of matrices for 22:for 2x2 becausefor any dimension
10. In many problems:timeindependenttimedependente.g., Molecule in a radiation field:(orthonormal)(time dependent phase factors)Copyright – Michael D. Fayer, 2018
11. For this situation:time evolution of density matrix elements, Cij(t), depends only on time dependent interaction termSee derivation in book – and lecture slides.Like first steps in time dependent perturbation theory before any approximations.In absence of , only time dependence from time dependent phase factorsfrom . No changes in magnitudes of coefficients Cij .Copyright – Michael D. Fayer, 2018
12. Time Dependent Two State Problem Revisited:Previously treated in Chapter 8 with Schrödinger Equation.Because degenerate states, time dependent phase factors cancel in off-diagonal matrix elements – special case.In general, the off-diagonal elements have time dependent phase factors.Copyright – Michael D. Fayer, 2018
13. UseMultiplying matrices and subtracting givesEquations of motion of density matrix elements:ProbabilitiesCoherencesCopyright – Michael D. Fayer, 2018
14. UsingTake time derivativeUsing Tr = 1, i.e., andSame result as Chapter 8 except obtained probabilities directly. No probability amplitudes.For initial condition at t = 0. Copyright – Michael D. Fayer, 2018
15. Can get off-diagonal elementsSubstituting:Copyright – Michael D. Fayer, 2018
16. ij density matrix elementDensity matrix elements have no time dependent phase factors.time dependent phase factor in ket, butits complex conjugate is in bra. Productis 1. Kets and bras normalized, closedbracket gives 1.Time dependent coefficient, but no phase factors.Copyright – Michael D. Fayer, 2018
17. Can be time dependent phase factors in density matrix equation of motion.s – spatialno time dependent phasefactortime dependent phase factorif E1 ≠ E2.Therefore, in general, the commutator matrix,will have time dependent phase factors if E1 ≠ E2.For two levels, but the same in any dimension.when you multiply it out,Copyright – Michael D. Fayer, 2018
18. Expectation Value of an OperatorMatrix elements of ADerivation in book and see lecture slidesExpectation value of A is trace of the product of density matrix with the operator matrix .Important: carries time dependence of coefficients.Time dependent phase factors may occur in off-diagonal matrix elements of A.Copyright – Michael D. Fayer, 2018
19. Example: Average E for two state problemOnly need to calculate thediagonal matrix elements.Time dependent phase factors cancel because degenerate. Special case. In general have time dependent phase factors.EE = 0Copyright – Michael D. Fayer, 2018
20. Working with basis set of eigenkets of time independent piece of Hamiltonian,H0, the time dependence of the density matrix depends only on the timedependent piece of the Hamiltonian, HI.Total Hamiltoniantime independentUseas basis set.Proof that only need consider when working in basis set of eigenvectors of H0.Copyright – Michael D. Fayer, 2018
21. Time derivative of density operator (using chain rule)(A)Use Schrödinger Equation and its complex conjugate(B)Substitute expansion into derivative terms in eq. (A).(C)(B) = (C)Copyright – Michael D. Fayer, 2018
22. Using Schrödinger EquationRight multiply top eq. by .Left multiply bottom equation by .GivesUsing these see that the 1st and 3rd termsin (B) cancel the 2nd and 4th terms in (C).(B)(C)Copyright – Michael D. Fayer, 2018
23. After canceling terms, (B) = (C) becomesConsider the ij matrix element of this expression.The matrix elements of the left hand side areIn the basis set of the eigenvectors of H0,H0 cancels out of equation of motion of density matrix.Copyright – Michael D. Fayer, 2018
24. Expectation valuecomplete orthonormal basis set.Matrix elements of AProof that =Copyright – Michael D. Fayer, 2018
25. note orderThenMatrix multiplication, Chapter 13 (13.18)like matrix multiplication but only diagonal elements – j on both sides.Also, double sum. Sum over j – sum diagonal elements.Therefore,Copyright – Michael D. Fayer, 2018
26. radiationfieldCoherent Coupling by of Energy Levels by Radiation FieldTwo state problemIn general, if radiation field frequency is near E, and other transitionsare far off resonance, can treat as a 2 state system.NMR – 2 spin states, magnetic transition dipoleCopyright – Michael D. Fayer, 2018
27. Molecular Eigenstates as BasisInteraction due to application of optical field (light) on or near resonance.Copyright – Michael D. Fayer, 2018
28. Take real (doesn’t change results)Define Rabi Frequency, 1Then is value of transition dipole bracket,Note – time independent kets. No phase factors. Have taken phase factors out.take out time dependent phase factorsCopyright – Michael D. Fayer, 2018
29. Blue diagonalRed off-diagonalGeneral state of systemCopyright – Michael D. Fayer, 2018
30. Equations of Motion of Density Matrix ElementsTreatment exact to this point (expect for dipole approx. in ).Copyright – Michael D. Fayer, 2018
31. Rotating Wave ApproximationPut this into equations of motionWill have terms likeTerms with off resonance Don’t cause transitionsLooks like high frequency Stark Effect Bloch – Siegert ShiftSmall but sometimes measurable shift in energy.Drop these terms!Copyright – Michael D. Fayer, 2018
32. With Rotating Wave ApproximationEquations of motion of density matrixThese are theOptical Bloch Equations for optical transitionsor just the Bloch Equations for NMR.Copyright – Michael D. Fayer, 2018H1 – oscillating magnetic field of applied RF.m – magnetic transition dipole.
33. Consider on resonance case = 0 Equations reduce toThese are IDENTICAL to the degenerate time dependent 2 state problemwith = 1/2. All of the phase factors = 1.Copyright – Michael D. Fayer, 2018
34. On resonance coupling to time dependent radiation field induces transitions.Looks identical to time independentcoupling of two degenerate states.In effect, the on resonance radiation field “removes” energy differences andtime dependence of field.Thenpopulationscoherencesat t = 0.Copyright – Michael D. Fayer, 2018
35. RecallThis is called a pulse inversion, all population in excited state.populationsThis is called a /2 pulse Maximizes off diagonal elements 12, 21As t is increased, populationoscillates between ground and excited state at Rabi frequency.Transient NutationCoherent Couplingtr22 – excited state prob.p pulse2p pulseCopyright – Michael D. Fayer, 2018
36. Off Resonance Coherent CouplingDefineFor same initial conditions:Solutions of Optical Bloch EquationsOscillations Faster eMax excited state probability:(Like non-degenerate time dependent 2-state problem)w1 = mE0 - Rabi frequencyAmount radiation field frequency is off resonance from transition frequency.Copyright – Michael D. Fayer, 2018
37. Near Resonance Case - ImportantThen11, 22 reduce to on resonance case.Same as resonance case except for phase factorThis is the basis of Fourier Transform NMR. Although spins havedifferent chemical shifts, make ω1 big enough, all look like on resonance.For /2 pulse, maximizes 12, 21 1t = /2 t << /2 0 ButThen, 12, 21 virtually identical to on resonance case and 11, 22 same as on resonance case.becauseCopyright – Michael D. Fayer, 2018
38. Free PrecessionAfter pulse of = 1t (flip angle)On or near resonance immediately after the pulse (t = 0)After pulse – no radiation field.Hamiltonian is H0 Copyright – Michael D. Fayer, 2018
39. Solutions11 = a constant = 11(0) 22 = a constant = 22(0) t = 0 is at end of pulseOff-diagonal density matrix elements Only time dependent phase factorPopulations don’t change.Copyright – Michael D. Fayer, 2018
40. Off-diagonal density matrix elements after pulse ends (t = 0).Consider expectation value of transition dipole .No time dependent phase factors.Phase factors were taken out of as partof the derivation. Matrix elementsinvolve time independent kets. t = 0, end of pulseCopyright – Michael D. Fayer, 2018
41. After pulse of = 1t (flip angle)On or near resonanceCopyright – Michael D. Fayer, 2018
42. Oscillating electric dipole (magnetic dipole - NMR) at frequency 0, Oscillating E-field (magnet field) Free precession.Rot. wave approx.Tip of vector goes in circle.Copyright – Michael D. Fayer, 2018
43. Pure and Mixed Density MatrixUp to this point - pure density matrix. One system or many identical systems.Mixed density matrix Describes nature of a collection of sub-ensembles each with different properties.The subensembles are not interacting.andPk probability of having kth sub-ensemble with density matrix, k.Density matrix for mixed systemsor integral if continuous distributionSum of probabilities (or integral) is unity.Total density matrix is the sum of the individual density matrices times their probabilities.Because density matrix is at probability level, can sum (see Errata and Addenda).Copyright – Michael D. Fayer, 2018
44. Example: Light coupled to two different transitions – free precessionDifference of both 01 & 02 from small compared to 1, that is, both near resonance.Equal probabilities P1=0.5 and P2=0.5For a given pulse of radiation field, both sub-ensembles will have same flip angle .CalculateLight frequency near 01 & 02.Copyright – Michael D. Fayer, 2018
45. Pure density matrix result for flip angle :For 2 transitions - P1=0.5 and P2=0.5from trig.identitiesCall: center frequency 0, shift from the center then, 01 = 0 + and 02 = 0 - , with << 0 Therefore,Beat gives transition frequencies – FT-NMRhigh freq. oscillation low freq. oscillation, beatCopyright – Michael D. Fayer, 2018
46. Equal amplitudes – 100% modulation, ω01 = 20.5; ω01 = 19.5Copyright – Michael D. Fayer, 2018
47. Amplitudes 2:1 – not 100% modulation , ω01 = 20.5; ω01 = 19.5Copyright – Michael D. Fayer, 2018
48. Amplitudes 9:1 – not 100% modulation , ω01 = 20.5; ω01 = 19.5Copyright – Michael D. Fayer, 2018
49. Equal amplitudes – 100% modulation, ω01 = 21; ω01 = 19Copyright – Michael D. Fayer, 2018
50. Free Induction Decaycenter freq0h frequencyof particular moleculew Frequently, distribution is a Gaussian - probability of finding a molecule at a particular frequency, Ph.Identical molecules haverange of transition frequencies.Different solvent environments.Doppler shifts, etc.Gaussian envelopestandard deviationnormalizationconstantThenpure density matrixprobability, Ph Copyright – Michael D. Fayer, 2018
51. Radiation field at = 0 line center 1 >> – all transitions near resonanceApply pulse with flip angle , transition dipole expectation value.Using result for single frequency h and flip angle Following pulse, each sub-ensemble will undergo free precession at hCopyright – Michael D. Fayer, 2018
52. Substituting = (h – 0), frequency of a molecule as difference from center frequency (light frequency).Then h = ( +0) and dωh = d.Oscillation at 0; decaying amplitude Gaussian decay with standard deviation in time 1/ (Free Induction Decay)Phase relationships lost Coherent Emission DecaysOff-diagonal density matrix elements – coherence; diagonal - magnitudeFirst integral zero; integral of an even function multiplying an odd function.With the trig identity:Copyright – Michael D. Fayer, 2018
53. flip angle light frequency free induction decayDecay of oscillating macroscopic dipole.Free induction decay.Coherent emissionof light.rotating frame at center freq., w0higher frequencieslower frequenciest = 0t = t'Copyright – Michael D. Fayer, 2018