5.76 Lecture #33 5/08/91 Page 1 of 10 pages. Lecture #33: Vibronic Coupling
|
|
- Brandon Edwards
- 5 years ago
- Views:
Transcription
1 5.76 Lecture #33 5/8/9 Page of pages Lecture #33: Vibroic Couplig Last time: H CO A A X A Electroically forbidde if A -state is plaar vibroically allowed to alterate v if A -state is plaar iertial defect says A -state is ot plaar expect to see all v if ot plaar staggerig of v level spacigs iversio through low barrier to plaarity dyamic vs. rigid molecule symmetry classificatio: molecular symmetry group How does vibroic couplig really work? What are the vibratioal itesity factors aalogous to Frack-Codo factors i the case of vibroically allowed rather tha electroically allowed trasitio? See T. Azumi ad K. Matsuzaki, Photochemistry ad Photobiology 5, 35 (977) for readable review article. Outlie: Crude Adiabatic Approximatio Correctio of ψ for effect of eglected off-diagoal matrix elemets H CO A A example What happes to Frack-Codo factors for vibroically allowed trasitio? Two electroic basis sets prediagoalize symmetry-breakig vibroic iteractio Chages i shapes of potetial curves Ies model for vibratioal bad itesities ad level staggerig Recall or-oppeheimer or clamped uclei approximatio. We use this procedure to defie complete sets of electroic ad uclear motio wavefuctios with which we ca FORMALLY expad exact ψ s ad compute (or parametrize) all properties of exact eigestates. The simplest basis set is called CRUDE ADIAATIC (CA) ψ CA ( r,q)=ψ o j ( r,q )χ CA fixed uc lear locatios! ( Q) Q is a coveiet referece structure (usually the equilibrium geometry or a high-symmetry potetial eergy maximum or saddle poit). ψ j o is the electroic wavefuctio i the j-th electroic state computed at the fixed uclear coordiates Q.
2 5.76 Lecture #33 5/8/9 Page of pages χ CA (Q) is the vibratio-rotatio wavefuctio computed from a Schrödiger Equatio. uclear kietic eergy potetial eergy of bare uclei eigevalue of clamped uclei electroic Schrödiger Equatio at Q approximate uclear U(r,Q) = U(r,Q) U(r,Q ) chage i e uclear ad e e Coulomb eergy T N (Q) + V(Q) + o j ( Q )+ ψ o j ( r,q )² U(r,Q) ψ o j r,q ( ) χ CA ( Q) effective potetialeergy surface CA = E χ CA ( Q) Note that the U itegral is evaluated usig effect of distortio of molecule from Q. ψ j o (r, Q ) thus caot cotai the exact We have explicitly excluded the effects of off-diagoal matrix elemets. I order to get a better approximatio to the exact ψ, we must use perturbatio theory to correct ψ. ψ (r,q) =ψ CA (r,q)+ kr =ψ o j (r,q )χ CA (Q) + ψ CA ²Uψ CA { } kr ψ kr E CA CA E kr k j ψ o k (r,q )χ CA kr (Q) r CA (r,q) ( χ CA kr ψ o o CA k ²Uψ j χ ) E CA CA E kr call this a vibroic { } meas itegrate over r ad Q ( ) meas itegrate over Q meas itegrate over r This form of ψ (r,q) is called the Herzberg-Teller expasio. Now expad U(r,Q) i power series about Q i each of the ormal coordiates.
3 5.76 Lecture #33 5/8/9 Page 3 of pages ²U= ² U(r,Q )+ U(r,Q) Q = by defiitio of U Now defie the mixig coefficiet. Q +,m U Q Q m Q Q m etc. γ kr, ( ) U ψ k o r,q Q ψ o j ( r,q ) χ CA CA kt Q χ E CA CA E kt ψ (r,q) =ψ o j (r,q )γ CA (Q) + ut we ca see that γ kr, must vaish if k j r γ kr, γ o k (r,q )χ CA kr (Q) ote vibroic wavefuctio for k-th, NOT j-th electroic state! Γ k Γ j Γ Q OR Γ r Γ t Γ Q which is equivalet to requirig that Γ kr Γ Γ totally symmetric. So ow we are ready to cosider the specific case of the H CO A A state. Out-of-plae edig mode as promoter b A = b vibratio So we are cosiderig vibroic couplig to the state. No-Lecture Let s make a really crude model for the out-of-plae bedig levels of both A ad states. * both are harmoic (N the A state is NOT a double miimum o-plaar state!!) * both have same frequecy ω * couplig is exclusively via U Q term. Q
4 5.76 Lecture #33 5/8/9 Page of pages 3a a b 5a b b b 6a X A ev elect. forbidde b b A A π* 3.5 ev elect. allowed (b-type) 6a b σ* 7. ev elect. allowed (a-type) b b A π* π 8. ev elect. allowed (c-type) b 5a π* σ 9.5 ev A X trasitio ca borrow oscillator stregth by vibroic couplig with via b vibratio because A b = A via a vibratio because A a = A a vibratio via b vibratio because A b = I will ow show that vibroic couplig accouts for both the oscillator stregth for A X trasitio ad the staggerig of ν vibratioal levels i A -state. Assume ν i A ad states is coveiet for harmoic - ot a double miimum o-plaar state calculatig same ω ad ot displaced (ecessary if miimum or maximum is plaar) vibratioal matrix elemets couplig is exclusively via ŽU Q term ŽQ
5 5.76 Lecture #33 5/8/9 Page 5 of pages v = A 3 v = ψ =ψχ + γ ψχ o CA o CA Av A Av v, Av v v o U v Q v o γ, ψ ψ v Av A CA Q E E Av a mass-idepedet electroic factor β A mode #, ot th power µω ( ) / v +δ CA v v, v + + v δ ( ) ² T A ω v v v,v Keepig oly levels of state i Herzberg- Teller expasio
6 5.76 Lecture #33 5/8/9 Page 6 of pages Summary of o-zero matrix elemets 3 v = 3 3 So we have 3 v = lump everythig ito this adjustable costat ψ =ψ χ +βψ v χ + v + χ ψ =ψ χ βψ v χ v + χ o CA o CA CA Av A Av v v + o CA o CA CA v v A Av Av +
7 state 5.76 Lecture #33 5/8/9 Page 7 of pages Trasitio probability for A v X v I = ψ µ ψ Av, X v Av X v µ v ( v ) o o / CA CA / CA CA b X v Xv v + Xv =β ψ ψ χ χ + + χ χ =β M b, X F-C factor vq ( v ) q v ( v ) positive v Xv v + Xv X X v, v v +, v / either sig Note that this is more complicated tha usual FRANCK-CONDON expressio for allowed trasitios. It is expressed i terms of Frack-Codo factors for X NOT A X!!!! We still have a symmetry selectio rule for the ν vibratioal modes because they are o-totally symmetric. From v = we ca oly reach v = eve or A v = odd. Note that the itesity expressio above vaishes for v = ad v = because q X,. To express this more geerally, for ay vibratioal bad i the A X system that is made allowed by vibroic couplig to the promoted by ν. I idividual mode F-C factors A X M q X v Q v,, v v AV XV b X v i vi vi v v symmetry selectio v rule v X = eve b-type v A v = odd v A v X = odd =β K. K. Ies J. Mol. Spectrosc. 99, 9 (983) performed a vibroic couplig calculatio which ot oly reproduced the mode- itesity promotio factors, but also explaied the level staggerig i the A -state.
8 5.76 Lecture #33 5/8/9 Page 8 of pages I order to defie complete basis sets, we solve a approximate Schrödiger equatio by eglectig certai terms i the exact H, or igorig certai off-diagoal elemets of certai terms. I the crude adiabatic approximatio, we defie potetial curves by igorig terms of the form ψ CA CA ² H(r,Q) ψ kr We showed that, by expadig U as power series i Q (the ormal mode displacemets), we get ( ) jk = H electroic ψ j o r,q Q ( ) U ( ) Q = ψ k o r,q γ jk Q We ca go to a ew electroic basis set by diagoalizig H + γ jk Q. Suppose we have two idetical harmoic potetial curves for mode of electroic states j ad k. The we have the followig zero-order ad diagoalized potetial curves. V k (Q ) V k o H =γ jk Q V j o
9 5.76 Lecture #33 5/8/9 Page 9 of pages Upper curve gets arrower. Lower curve turs ito double miimum curve. Q = poits of both curves do ot shift. Vibratioal Eigestates of lower curve will exhibit the patter of a symmetric double miimum potetial. V k o V j o H ij ( Q )=ω k Q ( Q )=ω j Q ( Q )=γ jk Q Secod-order perturbatio theory: ( γ jk ) Q V k =ω k Q + = ω ω ( k ω j )Q k Q +αq + T ek T ej V j = ω j Q αq ( ) ω ( k ω k )= ω ( j ω j )= γ jk T ek T ej opposite sig shifts i harmoic frequecy ( γ jk ) α ω ( T ek T ej ) k ( ω j ) quartic term that depeds o differece i ω's for j ad k. This shows that upper state ω icreases ad lower state ω decreases. Exact treatmet V ± = ω k +ω j ²V E H H ²V ( )Q ²V = ω k ω j Q ± ω ( k ω j ) E Q + γ jk V ± = V k + V j ( ) Q / ± ²V + H For large γ, secod term i [ ] / will domiate at small Q but first term will evetually domiate at large Q. /
10 5.76 Lecture #33 5/8/9 Page of pages A secod-order perturbatio treatmet of this kid of -state iteractio i the CA picture caot give this type of level stagger. It is ecessary to set up ad diagoalize two matrices H I odd quata of upper state eve quata of lower state H II eve quata of upper state odd quata of lower state because of odd-eve symmetry of a symmetric (ot ecessarily harmoic) potetial, there ca be o couplig matrix elemets betwee these two matrices. The level shifts are larger for the lower states i H II tha those i H I. This produces level staggerig. K. K. Ies [J. Mol. Spectrosc. 99, 9-3 (983)] reproduced A X itesity ad A -state level patter with T A A = 835 cm ω =ω = 5 cm, Av v= va A H =βv β= 338 / cm
5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpeCurseWare http://cw.mit.edu 5.8 Small-Mlecule Spectrscpy ad Dyamics Fall 8 Fr ifrmati abut citig these materials r ur Terms f Use, visit: http://cw.mit.edu/terms. 5.8 Lecture #33 Fall, 8 Page f
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpeCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy ad Dyamics Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture # 33 Supplemet
More information1 Adiabatic and diabatic representations
1 Adiabatic ad diabatic represetatios 1.1 Bor-Oppeheimer approximatio The time-idepedet Schrödiger equatio for both electroic ad uclear degrees of freedom is Ĥ Ψ(r, R) = E Ψ(r, R), (1) where the full molecular
More informationChapter 5 Vibrational Motion
Fall 4 Chapter 5 Vibratioal Motio... 65 Potetial Eergy Surfaces, Rotatios ad Vibratios... 65 Harmoic Oscillator... 67 Geeral Solutio for H.O.: Operator Techique... 68 Vibratioal Selectio Rules... 7 Polyatomic
More informationLecture 25 (Dec. 6, 2017)
Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed
More information1. Szabo & Ostlund: 2.1, 2.2, 2.4, 2.5, 2.7. These problems are fairly straightforward and I will not discuss them here.
Solutio set III.. Szabo & Ostlud:.,.,.,.5,.7. These problems are fairly straightforward ad I will ot discuss them here.. N! N! i= k= N! N! N! N! p p i j pi+ pj i j i j i= j= i= j= AA ˆˆ= ( ) Pˆ ( ) Pˆ
More informationHE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples:
5.6 4 Lecture #3-4 page HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT Do t restrict the wavefuctio to a sigle term! Could be a liear combiatio of several wavefuctios e.g. two terms:
More informationHydrogen (atoms, molecules) in external fields. Static electric and magnetic fields Oscyllating electromagnetic fields
Hydroge (atoms, molecules) i exteral fields Static electric ad magetic fields Oscyllatig electromagetic fields Everythig said up to ow has to be modified more or less strogly if we cosider atoms (ad ios)
More informationLECTURE 14. Non-linear transverse motion. Non-linear transverse motion
LETURE 4 No-liear trasverse motio Floquet trasformatio Harmoic aalysis-oe dimesioal resoaces Two-dimesioal resoaces No-liear trasverse motio No-liear field terms i the trajectory equatio: Trajectory equatio
More informationOffice: JILA A709; Phone ;
Office: JILA A709; Phoe 303-49-7841; email: weberjm@jila.colorado.edu Problem Set 5 To be retured before the ed of class o Wedesday, September 3, 015 (give to me i perso or slide uder office door). 1.
More informationPerturbation Theory I (See CTDL , ) Last time: derivation of all matrix elements for Harmonic-Oscillator: x, p, H. n ij n.
Perturbatio Theory I (See CTDL 1095-1107, 1110-1119) 14-1 Last time: derivatio of all matrix elemets for Harmoic-Oscillator: x, p, H selectio rules scalig ij x i j i steps of 2 e.g. x : = ± 3, ± 1 xii
More informationForces: Calculating Them, and Using Them Shobhana Narasimhan JNCASR, Bangalore, India
Forces: Calculatig Them, ad Usig Them Shobhaa Narasimha JNCASR, Bagalore, Idia shobhaa@jcasr.ac.i Shobhaa Narasimha, JNCASR 1 Outlie Forces ad the Hellma-Feyma Theorem Stress Techiques for miimizig a fuctio
More informationPhysics 232 Gauge invariance of the magnetic susceptibilty
Physics 232 Gauge ivariace of the magetic susceptibilty Peter Youg (Dated: Jauary 16, 2006) I. INTRODUCTION We have see i class that the followig additioal terms appear i the Hamiltoia o addig a magetic
More informationThe Born-Oppenheimer approximation
The Bor-Oppeheimer approximatio 1 Re-writig the Schrödiger equatio We will begi from the full time-idepedet Schrödiger equatio for the eigestates of a molecular system: [ P 2 + ( Pm 2 + e2 1 1 2m 2m m
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationTIME-CORRELATION FUNCTIONS
p. 8 TIME-CORRELATION FUNCTIONS Time-correlatio fuctios are a effective way of represetig the dyamics of a system. They provide a statistical descriptio of the time-evolutio of a variable for a esemble
More informationThe time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m "2 + V ( r,t) (1.
Adrei Tokmakoff, MIT Departmet of Chemistry, 2/13/2007 1-1 574 TIME-DEPENDENT QUANTUM MECHANICS 1 INTRODUCTION 11 Time-evolutio for time-idepedet Hamiltoias The time evolutio of the state of a quatum system
More information1. Hydrogen Atom: 3p State
7633A QUANTUM MECHANICS I - solutio set - autum. Hydroge Atom: 3p State Let us assume that a hydroge atom is i a 3p state. Show that the radial part of its wave fuctio is r u 3(r) = 4 8 6 e r 3 r(6 r).
More informationPHY4905: Nearly-Free Electron Model (NFE)
PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates
More informationOn the Quantum Theory of Molecules
O the Quatum Theory of Molecules M. Bor a, J.R. Oppeheimer b a Istitute of Theoretical Physics, Göttige b Istitute of Theoretical Physics, Göttige Abstract It will be show that the familiar compoets of
More informationLecture #18
18-1 Variatioal Method (See CTDL 1148-1155, [Variatioal Method] 252-263, 295-307[Desity Matrices]) Last time: Quasi-Degeeracy Diagoalize a part of ifiite H * sub-matrix : H (0) + H (1) * correctios for
More informationChapter 8 Approximation Methods, Hueckel Theory
Witer 3 Chem 356: Itroductory Quatum Mechaics Chapter 8 Approximatio Methods, Huecel Theory... 8 Approximatio Methods... 8 The Liear Variatioal Priciple... Chapter 8 Approximatio Methods, Huecel Theory
More informationVibrational Spectroscopy 1
Applied Spectroscopy Vibratioal Spectroscopy Recommeded Readig: Bawell ad McCash Chapter 3 Atkis Physical Chemistry Chapter 6 Itroductio What is it? Vibratioal spectroscopy detects trasitios betwee the
More information5.61 Fall 2013 Problem Set #3
5.61 Fall 013 Problem Set #3 1. A. McQuarrie, page 10, #3-3. B. McQuarrie, page 10, #3-4. C. McQuarrie, page 18, #4-11.. McQuarrie, pages 11-1, #3-11. 3. A. McQuarrie, page 13, #3-17. B. McQuarrie, page
More informationThere are 7 crystal systems and 14 Bravais lattices in 3 dimensions.
EXAM IN OURSE TFY40 Solid State Physics Moday 0. May 0 Time: 9.00.00 DRAFT OF SOLUTION Problem (0%) Itroductory Questios a) () Primitive uit cell: The miimum volume cell which will fill all space (without
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationPhysics 7440, Solutions to Problem Set # 8
Physics 7440, Solutios to Problem Set # 8. Ashcroft & Mermi. For both parts of this problem, the costat offset of the eergy, ad also the locatio of the miimum at k 0, have o effect. Therefore we work with
More informationSimilarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle
Similarity betwee quatum mechaics ad thermodyamics: Etropy, temperature, ad Carot cycle Sumiyoshi Abe 1,,3 ad Shiji Okuyama 1 1 Departmet of Physical Egieerig, Mie Uiversity, Mie 514-8507, Japa Istitut
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More informationQuantum Annealing for Heisenberg Spin Chains
LA-UR # - Quatum Aealig for Heiseberg Spi Chais G.P. Berma, V.N. Gorshkov,, ad V.I.Tsifriovich Theoretical Divisio, Los Alamos Natioal Laboratory, Los Alamos, NM Istitute of Physics, Natioal Academy of
More informationOn the Quantum Theory of Molecules M Born and J R Oppenheimer Ann. Physik 84, 457 (1927) January 23, 2002 Translated by S M Blinder with emendations b
O the Quatum Theory of Molecules M Bor ad J R Oppeheimer A. Physik 84, 457 (197) Jauary 3, Traslated by S M Blider with emedatios by Bria Sutclie ad Wolf Geppert Abstract It will be show that the familiar
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More information5.74 TIME-DEPENDENT QUANTUM MECHANICS
p. 1 5.74 TIME-DEPENDENT QUANTUM MECHANICS The time evolutio of the state of a system is described by the time-depedet Schrödiger equatio (TDSE): i t ψ( r, t)= H ˆ ψ( r, t) Most of what you have previously
More informationProblem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:
2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium
More informationTrue Nature of Potential Energy of a Hydrogen Atom
True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial
More informationQuantum Theory Assignment 3
Quatum Theory Assigmet 3 Assigmet 3.1 1. Cosider a spi-1/ system i a magetic field i the z-directio. The Hamiltoia is give by: ) eb H = S z = ωs z. mc a) Fid the Heiseberg operators S x t), S y t), ad
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationLecture 14 and 15: Algebraic approach to the SHO. 1 Algebraic Solution of the Oscillator 1. 2 Operator manipulation and the spectrum 4
Lecture 14 ad 15: Algebraic approach to the SHO B. Zwiebach April 5, 2016 Cotets 1 Algebraic Solutio of the Oscillator 1 2 Operator maipulatio ad the spectrum 4 1 Algebraic Solutio of the Oscillator We
More informationChemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics
Chemical Kietics CHAPTER 14 Chemistry: The Molecular Nature of Matter, 6 th editio By Jesperso, Brady, & Hyslop CHAPTER 14 Chemical Kietics Learig Objectives: Factors Affectig Reactio Rate: o Cocetratio
More informationAtomic Physics 4. Name: Date: 1. The de Broglie wavelength associated with a car moving with a speed of 20 m s 1 is of the order of. A m.
Name: Date: Atomic Pysics 4 1. Te de Broglie wavelegt associated wit a car movig wit a speed of 0 m s 1 is of te order of A. 10 38 m. B. 10 4 m. C. 10 4 m. D. 10 38 m.. Te diagram below sows tree eergy
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationAME 513. " Lecture 3 Chemical thermodynamics I 2 nd Law
AME 513 Priciples of Combustio " Lecture 3 Chemical thermodyamics I 2 d Law Outlie" Why do we eed to ivoke chemical equilibrium? Degrees Of Reactio Freedom (DORFs) Coservatio of atoms Secod Law of Thermodyamics
More informationRadiative Lifetimes of Rydberg States in Neutral Gallium
Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 3 Radiative Lifetimes of Rydberg States i Neutral Gallium M. Yildiz, G. Çelik ad H.Ş. Kiliç Departmet of Physics, Faculty of Arts ad Sciece, Selçuk Uiversity,
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationPHYS-505 Parity and other Discrete Symmetries Lecture-7!
PHYS-505 Parity ad other Discrete Symmetries Lecture-7! 1 Discrete Symmetries So far we have cosidered cotiuous symmetry operators that is, operatios that ca be obtaied by applyig successively ifiitesimal
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationNotes The Incremental Motion Model:
The Icremetal Motio Model: The Jacobia Matrix I the forward kiematics model, we saw that it was possible to relate joit agles θ, to the cofiguratio of the robot ed effector T I this sectio, we will see
More informationmx bx kx F t. dt IR I LI V t, Q LQ RQ V t,
Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationMIT Department of Chemistry 5.74, Spring 2005: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff
MIT Departmet of Chemistry 5.74, Sprig 5: Itroductory Quatum Mechaics II Istructor: Professor Adrei Tomaoff p. 97 ABSORPTION SPECTRA OF MOLECULAR AGGREGATES The absorptio spectra of periodic arrays of
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationPerturbation Theory, Zeeman Effect, Stark Effect
Chapter 8 Perturbatio Theory, Zeea Effect, Stark Effect Ufortuately, apart fro a few siple exaples, the Schrödiger equatio is geerally ot exactly solvable ad we therefore have to rely upo approxiative
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig
More informationChapter 14: Chemical Equilibrium
hapter 14: hemical Equilibrium 46 hapter 14: hemical Equilibrium Sectio 14.1: Itroductio to hemical Equilibrium hemical equilibrium is the state where the cocetratios of all reactats ad products remai
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationNotes for Lecture 5. 1 Grover Search. 1.1 The Setting. 1.2 Motivation. Lecture 5 (September 26, 2018)
COS 597A: Quatum Cryptography Lecture 5 (September 6, 08) Lecturer: Mark Zhadry Priceto Uiversity Scribe: Fermi Ma Notes for Lecture 5 Today we ll move o from the slightly cotrived applicatios of quatum
More informationR is a scalar defined as follows:
Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationPATH INTEGRAL for the HARMONIC OSCILLATOR
PATH ITEGRAL for the HAROIC OSCILLATOR I class, I have showed how to use path itegral formalism to calculate the partitio fuctio of a quatum system. Formally, ] ZT = Tr e it Ĥ = ÛT, 0 = dx 0 Ux 0, T ;
More informationPhysics Supplement to my class. Kinetic Theory
Physics Supplemet to my class Leaers should ote that I have used symbols for geometrical figures ad abbreviatios through out the documet. Kietic Theory 1 Most Probable, Mea ad RMS Speed of Gas Molecules
More informationPhysics 2D Lecture Slides Lecture 22: Feb 22nd 2005
Physics D Lecture Slides Lecture : Feb d 005 Vivek Sharma UCSD Physics Itroducig the Schrodiger Equatio! (, t) (, t) #! " + U ( ) "(, t) = i!!" m!! t U() = characteristic Potetial of the system Differet
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationFluid Physics 8.292J/12.330J % (1)
Fluid Physics 89J/133J Problem Set 5 Solutios 1 Cosider the flow of a Euler fluid i the x directio give by for y > d U = U y 1 d for y d U + y 1 d for y < This flow does ot vary i x or i z Determie the
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Uiversity of Wasigto Departmet of Cemistry Cemistry 453 Witer Quarter 15 Lecture 14. /11/15 Recommeded Text Readig: Atkis DePaula: 9.1, 9., 9.3 A. Te Equipartitio Priciple & Eergy Quatizatio Te Equipartio
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More informationGuiding-center transformation 1. δf = q ( c v f 0, (1) mc (v B 0) v f 0. (3)
Guidig-ceter trasformatio 1 This is my otes whe readig Liu Che s book[1]. 1 Vlasov equatio The liearized Vlasov equatio is [ t + v x + q m E + v B c ] δf = q v m δe + v δb c v f, 1 where f ad δf are the
More informationCMSE 820: Math. Foundations of Data Sci.
Lecture 17 8.4 Weighted path graphs Take from [10, Lecture 3] As alluded to at the ed of the previous sectio, we ow aalyze weighted path graphs. To that ed, we prove the followig: Theorem 6 (Fiedler).
More informationPATH INTEGRAL for HARMONIC OSCILLATOR
PATH ITEGRAL for HAROIC OSCILLATOR I class, I have showed how to use path itegral formalism to calculate the partitio fuctio of a quatum system. Formally, ] ZT = Tr e it Ĥ = ÛT, 0 = dx 0 Ux 0, T ; x 0,
More informationChapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io
Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More informationLecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box
561 Fall 013 Lecture #5 page 1 Last time: Lecture #5: Begi Quatum Mechaics: Free Particle ad Particle i a 1D Box u 1 u 1-D Wave equatio = x v t * u(x,t): displacemets as fuctio of x,t * d -order: solutio
More informationpoint, requiring all 4 curves to be continuous Discontinuity in P
. Solutio Methods a Classical approach Basic equatios of stellar structure + boudary coditios i Classical (historical method of solutio: Outward itegratio from ceter, where l = m = ward itegratio from
More informationBalancing. Rotating Components Examples of rotating components in a mechanism or a machine. (a)
alacig NOT COMPLETE Rotatig Compoets Examples of rotatig compoets i a mechaism or a machie. Figure 1: Examples of rotatig compoets: camshaft; crakshaft Sigle-Plae (Static) alace Cosider a rotatig shaft
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationI. ELECTRONS IN A LATTICE. A. Degenerate perturbation theory
1 I. ELECTRONS IN A LATTICE A. Degeerate perturbatio theory To carry out a degeerate perturbatio theory calculatio we eed to cocetrate oly o the part of the Hilbert space that is spaed by the degeerate
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More information( ) ( ), (S3) ( ). (S4)
Ultrasesitivity i phosphorylatio-dephosphorylatio cycles with little substrate: Supportig Iformatio Bruo M.C. Martis, eter S. Swai 1. Derivatio of the equatios associated with the mai model From the differetial
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More information10. Second quantization: molecule-radiation interaction
10. Secod quatizatio: molecule-radiatio iteractio Now that the full molecular (sectios 7 through 9), eld (b), ad iteractio (b) Hamiltoia operators are i had, they ca be combied to yield the overall molecule-radiatio
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationCOURSE INTRODUCTION: WHAT HAPPENS TO A QUANTUM PARTICLE ON A PENDULUM π 2 SECONDS AFTER IT IS TOSSED IN?
COURSE INTRODUCTION: WHAT HAPPENS TO A QUANTUM PARTICLE ON A PENDULUM π SECONDS AFTER IT IS TOSSED IN? DROR BAR-NATAN Follows a lecture give by the author i the trivial otios semiar i Harvard o April 9,
More information2C09 Design for seismic and climate changes
2C09 Desig for seismic ad climate chages Lecture 02: Dyamic respose of sigle-degree-of-freedom systems I Daiel Grecea, Politehica Uiversity of Timisoara 10/03/2014 Europea Erasmus Mudus Master Course Sustaiable
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More information