8.3 Perturbation theory

Size: px
Start display at page:

Download "8.3 Perturbation theory"

Transcription

1 8.3 Perturbatio theory Slides: Video Costructig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (u to First order erturbatio theory )

2 Perturbatio theory Costructig erturbatio theory Quatu echaics for scietists ad egieers David Miller

3 Tie-ideedet erturbatio theory Presue soe uerturbed Hailtoia H that has kow oralized eige solutios i.e., H 0 We ca iagie that our erturbatio could be rogressively tured o at least i a atheatical sese For exale we could be rogressively icreasig alied field fro zero to soe secific value o

4 Tie-ideedet erturbatio theory We look successively for the chages i the solutios for exale, for the th eergy eigevalue roortioal first to electric field first-order correctios (0) a (0) 0 lectric field,

5 Tie-ideedet erturbatio theory We look successively for the chages i the solutios for exale, for the th eergy eigevalue roortioal first to electric field first-order correctios roortioal to 2 secod-order correctios ad so o (0) 2 a b 0 (0) lectric field,

6 Tie-ideedet erturbatio theory It is ore coveiet ad geeral if we iagie a secific fixed erturbatio (e.g., a field ) ad we atheatically icrease a house-keeig araeter fro 0 to 1 so our erturbatio is with fixed Now we exress chages as orders of rather tha of the field itself (0) (1) 2 (2) (0) 0 1

7 The house-keeig araeter (0) 2 So, istead of writig ab (0) (1) 2 (2) we are writig ad istead of workig out a ad b we are goig to work out araeters (1) (2) ad ad so o These have diesios of eergy ad reflect the first order ad secod order correctios to the eergy as a result of the secific erturbatio e.g., a secific field

8 The house-keeig araeter I geeral, the, we iagie that our erturbed syste has soe additioal ter i the Hailtoia the erturbig Hailtoia H I our exale case of a ifiitely dee otetial well with a alied field that erturbig Hailtoia would be H e zl /2 I the theory, we write the erturbig Hailtoia as H usig to kee track of the order of the correctios through the owers of i the exressios We ca set = 1 at the ed if we like z

9 The house-keeig araeter (0) 2 So, we could set u the theory usig ab i which case we would work out a ad b ad soe other araeters But, to ake it ore geeral we use (0) (1) 2 (2) ad work out the araeters (1) (2) ad ad soe other araeters If this is cofusig at first the just thik of as the stregth of the electric field i our secific roble

10 Costructio of the orders of erturbatio theory With this way of thikig about the roble atheatically we ca write the erturbed Schrödiger equatio as H o H We ow resue that we ca exress the resultig erturbed eigefuctio ad eigevalue as ower series i this araeter, i.e.,

11 Costructio of the orders of erturbatio theory We ow substitute these ower series ito the erturbed Schrödiger equatio H o H to get H 0 H

12 Costructio of the orders of erturbatio theory Now, at ay secific oit i sace, each fuctio ad each fuctio H 0 H is just soe uber So, at ay secific oit i sace, the left had side of H H is just a ower series i, e.g., ad so is the right had side, e.g., a a a a b b b b

13 Costructio of the orders of erturbatio theory Because a ower series exasio is uique the oly way the equality of two ower series ca work a a a a b b b b for every value of withi soe covergece rage e.g., 0 to 1 is if the ters are equal, oe by oe, i.e., a0 b0 a1 b a 1 2 b2 a3 b3 ad so o

14 Costructio of the orders of erturbatio theory Hece, i H 0 H we ca equate each ter with a secific ower of ad hece obtai a rogressive set of equatios which we ca solve to evaluate correctios to whatever order we wish

15 Progressive set of erturbatio theory equatios I H 0 H equatig ters i 0, i.e., ters without gives the zeroth order equatio H o i.e., the uerturbed Hailtoia equatio with eigefuctios ad eigevalues So if we ow resue we start i a secific eigestate we write ad 0 istead of ad 0

16 Progressive set of erturbatio theory equatios So, with H 0 H we get a rogressive set of equatios each equatig a differet ower of H ad so o o H H o H H o

17 Progressive set of erturbatio theory equatios We ca rewrite these equatios as o H H H o H H o ad so o H 0 o 1 1 H H o H H o

18

19 8.3 Perturbatio theory Slides: Video First ad secod order theories Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (startig at First order erturbatio theory u to xale of well with field )

20 Perturbatio theory First ad secod order theories Quatu echaics for scietists ad egieers David Miller

21 First order erturbatio theory Now we ca calculate the various erturbatio ters 1 1 Startig with H H o ad reultilyig by gives H H 0 o o 1 H 1 H i.e., 1 H a forula for the first-order eergy correctio i the resece of our erturbatio H 1

22 First order erturbatio theory For the first order correctio to the wavefuctio we exad that correctio i the basis set 1 1 a Substitutig this is i 1 1 H H o i ad reultilyig by gives 1 1 H a 1 i o i i 1 i H i i 1 1 i i H

23 First order erturbatio theory So we have i ai i i H We resue the eergy eigevalue is ot degeerate i.e., oly oe eigefuctio for this eigevalue With o degeeracy, we still eed to distiguish two cases First, for, fro above 1 i i ai i H 1 i H So ai i Secod, for i 1 1 a 0a H (1) a which gives o costraits o 1 1 0

24 First order erturbatio theory (1) We are therefore free to choose The choice that akes the algebra silest 1 is to set a 0 which is the sae as sayig 1 we choose to ake orthogoal to The sae haes for the higher order equatios Hece, quite geerally we ake the coveiet choice j 0 a

25 First order erturbatio theory 1 i H 1 Hece with ai ad a 0 i the first order correctio to the wavefuctio is 1 H ad we have the first order correctio to the eergy 1 H

26 This iage caot curretly be dislayed. Secod order erturbatio theory We cotiue siilarly to fid the higher order ters Preultilyig H 2 1 H 1 2 o both sides by so o H gives 0 H H j Sice we chose orthogoal to H 2 1

27 Secod order erturbatio theory Usig our result for the first-order wavefuctio correctio 1 H the fro we obtai quivaletly H H H H 2

28 Secod order erturbatio theory For the secod order wavefuctio correctio 2 we exad 2 otig ow that is chose orthogoal to a 2 2 We reultily H 2 1 H 1 2 by i to obtai H 2 a 2 o i o i i H i i i i a a H

29 Secod order erturbatio theory So, we have a a a H i i i i Note this suatio excludes the ter = 1 because we chose to be orthogoal to 1 i.e., we have chose a 0 Hece, for i we have a i H ai ai i i Note that the secod order wavefuctio deeds oly o the first order eergy ad wavefuctio

30 First ad secod order erturbatio results 1 H First order a 1 1 a 1 i H i i, a 1 0 H 2 1 Secod order a 2 2 a 2 i a i H ai i i, a 2 0

31

32 8.3 Perturbatio theory Slides: Video Alyig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (startig at xale of well with field )

33 Perturbatio theory Alyig erturbatio theory Quatu echaics for scietists ad egieers David Miller

34 xale of a well with field We write the Hailtoia as the su of the uerturbed Hailtoia which is, i the well, i our diesioless uits 2 1 d H o 2 2 d ad the erturbig Hailtoia H f 1/2 where agai we take f 3 for a exlicit calculatio

35 First order eergy correctio I first order, the eergy shift with alied field is 2si 1/2 2si 1 H f 0 2 2f 1/2 si d 0 d The itegrals here are zero for all because the sie squared fuctio is eve with resect to the ceter of the well whereas 1/2 is odd Hece, for this articular roble there is o first order eergy correctio

36 First order eergy correctio There o first order eergy correctio because of syetry

37 First order eergy correctio There o first order eergy correctio because of syetry If the eergy chaged roortioately with alied field?

38 First order eergy correctio There o first order eergy correctio because of syetry If the eergy chaged roortioately with alied field chagig field directio (or sig) would chage the eergy correctio sig But, by syetry here the eergy chage caot deed field directio?

39 Matrix eleets for erturbatio calculatios The geeral atrix eleets that we will eed for further erturbatio calculatios are 1 0 H H 2si u 1/2 2si v d uv u v f I geeral we eed u ad v to have oosite arity i.e., if oe is odd, the other ust be eve for these atrix eleets to be o-zero sice otherwise the overall itegrad is odd about 1/2

40 First order correctio to the wavefuctio We calculate the first order wavefuctio correctio for the first state, i.e., for = 1 q i H 1 a ai 2 o 1 oi 2 where o are the eergies of the uerturbed states, ad q is a fiite uber we ust choose i ractice Here, we chose q = 6 though a saller uber would likely be quite accurate

41 First order correctio to the wavefuctio xlicitly, for the exasio coefficiets 1 a H / i i 1 o1 oi for 3 uits of field we have uerically 1 1 a a Here the value of for a 2 coares closely with the value of obtaied above i the fiite basis subset ethod 1 a

42 First order correctio to the wavefuctio We su the zero-order (uerturbed) wavefuctio si

43 First order correctio to the wavefuctio ad the first order correctio art fro the secod basis fuctio si 2

44 First order correctio to the wavefuctio To get our aroxiate wavefuctio solutio si si2

45 First order correctio to the wavefuctio Addig the ext correctio akes egligible differece si si si4

46 Secod order eergy correctio Sice the first order correctio to the eergy was zero to get a erturbatio correctio to the eergy we go to secod order 2 q 2 1 H 1 xlicitly, we have 1 H which uerically here gives or a total eergy of which coares with the result of fro the fiite basis subset ethod 2 2 1

47 Aroxiate aalytic forulas Note that is aalytically roortioal to the square of the field f q H /2 q f /2 q f

48 Aroxiate aalytic forulas Hece erturbatio theory gives a aroxiate aalytic result for the eergy which we ca ow use for ay field xlicitly, we ca write for the eergy of the first state i diesioless uits f This tyical kid of result fro erturbatio theory gives us a aroxiate aalytic forula valid for sall erturbatios

49 Aroxiate aalytic forulas Siilarly, for the wavefuctio the correctio is aroxiately roortioal to field for exale with exasio coefficiet 1 i H i 1/2 ai f i i So, keeig oly the doiat cotributio fro the secod-state wavefuctio i our exale we would have the aroxiate forula for sall f 2 si 0.06f 2 si 2 (This is ot quite oralized, though that could be doe)

50 Aroxiate aalytic results f Wavefuctio ergy Field

51

Perturbation Theory, Zeeman Effect, Stark Effect

Perturbation 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 information

X. Perturbation Theory

X. 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 information

Chapter 4 Postulates & General Principles of Quantum Mechanics

Chapter 4 Postulates & General Principles of Quantum Mechanics Chapter 4 Postulates & Geeral Priciples of Quatu Mechaics Backgroud: We have bee usig quite a few of these postulates already without realizig it. Now it is tie to forally itroduce the. State of a Syste

More information

( ) Time-Independent Perturbation Theory. Michael Fowler 2/16/06

( ) Time-Independent Perturbation Theory. Michael Fowler 2/16/06 Tie-Idepedet Perturbatio Theory Michael Fowler /6/6 Itroductio If a ato (ot ecessarily i its groud state) is placed i a exteral electric field, the eergy levels shift, ad the wave fuctios are distorted

More information

Physics 219 Summary of linear response theory

Physics 219 Summary of linear response theory 1 Physics 219 Suary of liear respose theory I. INTRODUCTION We apply a sall perturbatio of stregth f(t) which is switched o gradually ( adiabatically ) fro t =, i.e. the aplitude of the perturbatio grows

More information

Orthogonal Functions

Orthogonal Functions Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they

More information

Primes of the form n 2 + 1

Primes of the form n 2 + 1 Itroductio Ladau s Probles are four robles i Nuber Theory cocerig rie ubers: Goldbach s Cojecture: This cojecture states that every ositive eve iteger greater tha ca be exressed as the su of two (ot ecessarily

More information

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20 ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we

More information

6.3 Testing Series With Positive Terms

6.3 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 information

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3

More information

Complete Solutions to Supplementary Exercises on Infinite Series

Complete Solutions to Supplementary Exercises on Infinite Series Coplete Solutios to Suppleetary Eercises o Ifiite Series. (a) We eed to fid the su ito partial fractios gives By the cover up rule we have Therefore Let S S A / ad A B B. Covertig the suad / the by usig

More information

Lecture 25 (Dec. 6, 2017)

Lecture 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 information

Integrals of Functions of Several Variables

Integrals of Functions of Several Variables Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio

More information

AVERAGE MARKS SCALING

AVERAGE MARKS SCALING TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I

More information

GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog

GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog GENERALIZED KERNEL AND MIXED INTEGRAL EQUATION OF FREDHOLM - VOLTERRA TYPE R. T. Matoog Assistat Professor. Deartet of Matheatics, Faculty of Alied Scieces,U Al-Qura Uiversity, Makkah, Saudi Arabia Abstract:

More information

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1 ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like

More information

PHY4905: Nearly-Free Electron Model (NFE)

PHY4905: 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 information

Chapter 4. Fourier Series

Chapter 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

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia

More information

Birth-Death Processes. Outline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Relationship Among Stochastic Processes.

Birth-Death Processes. Outline. EEC 686/785 Modeling & Performance Evaluation of Computer Systems. Relationship Among Stochastic Processes. EEC 686/785 Modelig & Perforace Evaluatio of Couter Systes Lecture Webig Zhao Deartet of Electrical ad Couter Egieerig Clevelad State Uiversity webig@ieee.org based o Dr. Raj jai s lecture otes Relatioshi

More information

The Binomial Multi-Section Transformer

The Binomial Multi-Section Transformer 4/15/2010 The Bioial Multisectio Matchig Trasforer preset.doc 1/24 The Bioial Multi-Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where:

More information

x !1! + 1!2!

x !1! + 1!2! 4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio

More information

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) = AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,

More information

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

Orthogonal Function Solution of Differential Equations

Orthogonal Function Solution of Differential Equations Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,

More information

S. A. ALIEV, Y. I. YELEYKO, Y. V. ZHERNOVYI. STEADY-STATE DISTRIBUTIONS FOR CERTAIN MODIFICATIONS OF THE M/M/1/m QUEUEING SYSTEM

S. A. ALIEV, Y. I. YELEYKO, Y. V. ZHERNOVYI. STEADY-STATE DISTRIBUTIONS FOR CERTAIN MODIFICATIONS OF THE M/M/1/m QUEUEING SYSTEM Trasactios of Azerbaija Natioal Acadey of Scieces, Series of Physical-Techical ad Matheatical Scieces: Iforatics ad Cotrol Probles 009 Vol XXIX, 6 P 50-58 S A ALIEV, Y I YELEYKO, Y V ZHERNOVYI STEADY-STATE

More information

DIRICHLET CHARACTERS AND PRIMES IN ARITHMETIC PROGRESSIONS

DIRICHLET CHARACTERS AND PRIMES IN ARITHMETIC PROGRESSIONS DIRICHLET CHARACTERS AND PRIMES IN ARITHMETIC PROGRESSIONS We la to rove the followig Theore (Dirichlet s Theore) Let (a, k) = The the arithetic rogressio cotais ifiitely ay ries a + k : = 0,, 2, } = :

More information

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1)

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1) Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher

More information

10.6 ALTERNATING SERIES

10.6 ALTERNATING SERIES 0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

More information

1. C only. 3. none of them. 4. B only. 5. B and C. 6. all of them. 7. A and C. 8. A and B correct

1. C only. 3. none of them. 4. B only. 5. B and C. 6. all of them. 7. A and C. 8. A and B correct M408D (54690/54695/54700), Midterm # Solutios Note: Solutios to the multile-choice questios for each sectio are listed below. Due to radomizatio betwee sectios, exlaatios to a versio of each of the multile-choice

More information

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

Ratio of Two Random Variables: A Note on the Existence of its Moments

Ratio of Two Random Variables: A Note on the Existence of its Moments Metodološki zvezki, Vol. 3, o., 6, -7 Ratio of wo Rado Variables: A ote o the Existece of its Moets Ato Cedilik, Kataria Košel, ad Adre Bleec 3 Abstract o eable correct statistical iferece, the kowledge

More information

The Growth of Functions. Theoretical Supplement

The 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 information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm

More information

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k) THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

5.6 Binomial Multi-section Matching Transformer

5.6 Binomial Multi-section Matching Transformer 4/14/21 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-25 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply 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 information

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe

More information

Bertrand s postulate Chapter 2

Bertrand s postulate Chapter 2 Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture 9: Pricipal Compoet Aalysis The text i black outlies mai ideas to retai from the lecture. The text i blue give a deeper uderstadig of how we derive or get

More information

The Binomial Multi- Section Transformer

The Binomial Multi- Section Transformer 4/4/26 The Bioial Multisectio Matchig Trasforer /2 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: ( ω ) = + e +

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

PUTNAM TRAINING PROBABILITY

PUTNAM TRAINING PROBABILITY PUTNAM TRAINING PROBABILITY (Last udated: December, 207) Remark. This is a list of exercises o robability. Miguel A. Lerma Exercises. Prove that the umber of subsets of {, 2,..., } with odd cardiality

More information

5.6 Binomial Multi-section Matching Transformer

5.6 Binomial Multi-section Matching Transformer 4/14/2010 5_6 Bioial Multisectio Matchig Trasforers 1/1 5.6 Bioial Multi-sectio Matchig Trasforer Readig Assiget: pp. 246-250 Oe way to axiize badwidth is to costruct a ultisectio Γ f that is axially flat.

More information

Physics 232 Gauge invariance of the magnetic susceptibilty

Physics 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 information

42 Dependence and Bases

42 Dependence and Bases 42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V

More information

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi

More information

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

c z c z c z c z c c c c a a c A P( a ) a c a z c x z R j z z, z z z z I

c z c z c z c z c c c c a a c A P( a ) a c a z c x z R j z z, z z z z I . Quatu Spi States State Vectors c c i Ier Products: * * c c oralied. orthogoal c c c c c c * * i ii Probability that a particle i state ca be foud i state. c. States i S -basis y i i Geeral States. c

More information

Complex Numbers Solutions

Complex Numbers Solutions Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i

More information

HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples:

HE 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 information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES

PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with

More information

EXPERIMENTING WITH MAPLE TO OBTAIN SUMS OF BESSEL SERIES

EXPERIMENTING WITH MAPLE TO OBTAIN SUMS OF BESSEL SERIES EXPERIMENTING WITH MAPLE TO OBTAIN SUMS OF BESSEL SERIES Walter R Bloom Murdoch Uiversity Perth, Wester Australia Email: bloom@murdoch.edu.au Abstract I the study of ulse-width modulatio withi electrical

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Special Modeling Techniques

Special Modeling Techniques Colorado School of Mies CHEN43 Secial Modelig Techiques Secial Modelig Techiques Summary of Toics Deviatio Variables No-Liear Differetial Equatios 3 Liearizatio of ODEs for Aroximate Solutios 4 Coversio

More information

1 Approximating Integrals using Taylor Polynomials

1 Approximating Integrals using Taylor Polynomials Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................

More information

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 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 information

Lecture 20 - Wave Propagation Response

Lecture 20 - Wave Propagation Response .09/.093 Fiite Eleet Aalysis of Solids & Fluids I Fall 09 Lecture 0 - Wave Propagatio Respose Prof. K. J. Bathe MIT OpeCourseWare Quiz #: Closed book, 6 pages of otes, o calculators. Covers all aterials

More information

New Version of the Rayleigh Schrödinger Perturbation Theory: Examples

New Version of the Rayleigh Schrödinger Perturbation Theory: Examples New Versio of the Rayleigh Schrödiger Perturbatio Theory: Examples MILOŠ KALHOUS, 1 L. SKÁLA, 1 J. ZAMASTIL, 1 J. ČÍŽEK 2 1 Charles Uiversity, Faculty of Mathematics Physics, Ke Karlovu 3, 12116 Prague

More information

Lecture 19. Curve fitting I. 1 Introduction. 2 Fitting a constant to measured data

Lecture 19. Curve fitting I. 1 Introduction. 2 Fitting a constant to measured data Lecture 9 Curve fittig I Itroductio Suppose we are preseted with eight poits of easured data (x i, y j ). As show i Fig. o the left, we could represet the uderlyig fuctio of which these data are saples

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Chapter 9: Numerical Differentiation

Chapter 9: Numerical Differentiation 178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical

More information

Binomial transform of products

Binomial transform of products Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

Math 25 Solutions to practice problems

Math 25 Solutions to practice problems Math 5: Advaced Calculus UC Davis, Sprig 0 Math 5 Solutios to practice problems Questio For = 0,,, 3,... ad 0 k defie umbers C k C k =! k!( k)! (for k = 0 ad k = we defie C 0 = C = ). by = ( )... ( k +

More information

Random Models. Tusheng Zhang. February 14, 2013

Random Models. Tusheng Zhang. February 14, 2013 Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the

More information

Today in Physics 217: separation of variables IV

Today in Physics 217: separation of variables IV Today i Physics 27: separatio of variables IV Separatio i cylidrical coordiates Exaple of the split cylider (solutio sketched at right) More o orthogoality of trig fuctios ad Fourier s trick V = V V =

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

5.61 Fall 2013 Problem Set #3

5.61 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 information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

Problem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient

Problem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case

More information

Physics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing

Physics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals

More information

9.3 Power Series: Taylor & Maclaurin Series

9.3 Power Series: Taylor & Maclaurin Series 9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009 18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were

More information

Math 163 REVIEW EXAM 3: SOLUTIONS

Math 163 REVIEW EXAM 3: SOLUTIONS Math 63 REVIEW EXAM 3: SOLUTIONS These otes do ot iclude solutios to the Cocept Check o p8. They also do t cotai complete solutios to the True-False problems o those pages. Please go over these problems

More information

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

More information

Wavelet Transform Theory. Prof. Mark Fowler Department of Electrical Engineering State University of New York at Binghamton

Wavelet Transform Theory. Prof. Mark Fowler Department of Electrical Engineering State University of New York at Binghamton Wavelet Trasfor Theory Prof. Mark Fowler Departet of Electrical Egieerig State Uiversity of New York at Bighato What is a Wavelet Trasfor? Decopositio of a sigal ito costituet parts Note that there are

More information

Linear Regression Demystified

Linear 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 information

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Subject: Differential Equations & Mathematical Modeling-III

Subject: Differential Equations & Mathematical Modeling-III Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

Example of CLT for Symmetric Laminate with Mechanical Loading

Example of CLT for Symmetric Laminate with Mechanical Loading Exaple of CL for Syetric Laate with Mechaical Loadg hese are two probles which detail the process through which the laate deforatios are predicted usg Classical Laatio theory. he first oly cludes echaical

More information

Signals & Systems Chapter3

Signals & Systems Chapter3 Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of

More information

Lecture 11. Solution of Nonlinear Equations - III

Lecture 11. Solution of Nonlinear Equations - III Eiciecy o a ethod Lecture Solutio o Noliear Equatios - III The eiciecy ide o a iterative ethod is deied by / E r r: rate o covergece o the ethod : total uber o uctios ad derivative evaluatios at each step

More information

7 Sequences of real numbers

7 Sequences of real numbers 40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

More information

Different kinds of Mathematical Induction

Different kinds of Mathematical Induction Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}

More information

Question 1: The magnetic case

Question 1: The magnetic case September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to

More information

( ) GENERATING FUNCTIONS

( ) GENERATING FUNCTIONS GENERATING FUNCTIONS Solve a ifiite umber of related problems i oe swoop. *Code the problems, maipulate the code, the decode the aswer! Really a algebraic cocept but ca be eteded to aalytic basis for iterestig

More information

The Born-Oppenheimer approximation

The 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 information

MATH 1910 Workshop Solution

MATH 1910 Workshop Solution MATH 90 Workshop Solutio Fractals Itroductio: Fractals are atural pheomea or mathematical sets which exhibit (amog other properties) self similarity: o matter how much we zoom i, the structure remais the

More information

Define a Markov chain on {1,..., 6} with transition probability matrix P =

Define a Markov chain on {1,..., 6} with transition probability matrix P = Pla Group Work 0. The title says it all Next Tie: MCMC ad Geeral-state Markov Chais Midter Exa: Tuesday 8 March i class Hoework 4 due Thursday Uless otherwise oted, let X be a irreducible, aperiodic Markov

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

Math 475, Problem Set #12: Answers

Math 475, Problem Set #12: Answers Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe

More information