Quantum Physics III (8.06) Spring 2005 Solution Set 5

Size: px
Start display at page:

Download "Quantum Physics III (8.06) Spring 2005 Solution Set 5"

Transcription

1 Quntum Physics III (8.06 Spring 005 Solution Set 5 Mrch 8, 004. The frctionl quntum Hll effect (5 points As we increse the flux going through the solenoid, we increse the mgnetic field, nd thus the vector potentil is time-dependent. This in turn induces n electric field, the mgnitude of which is E φ = A φ c t = Φ Q t πrc. This electric field in turn induces current, j = t l = σe. Integrting this over φ, we find the chrge tht flows through the circulr pth: Q = σ Φ dφ r πrc = σφ 0 c. We lerned on the previous problem set tht R H = σ H, so using σ H = e /3h, we find Q = e 3. The frctionl chrge of the excittion leds to lot of very interesting physics; one notble effect is tht the qusiprticles behve neither s fermions or s bosons, but rther s something in between ( nyons.. A delt-function bump in the center of n infinite squre well (0 points ( ( point The eigenfunctions for the unperturbed Hmiltonin re ( πnx ψn(x 0 = cos n odd ( πnx = sin n even (b ( points The first order correction to the energies due to the perturbtion is given by / E n ( = n H n = dx ψn(x 0 αδ(x / = α ψ 0 n(0. Thus E n ( = α if n is odd, nd E( n = 0 if n is even. The energies re unchnged for even n becuse the wve functions ψk 0 vnish t x = 0, nd therefore do not feel the delt function perturbtion, which hs support only t tht point. (c (3 points The correction to the ground stte wve function is, to first order, ψ ( H m m ψ E m. 0 The mtrix element of the perturbtion is 0 E0 m { / α H n = dx ψ(xψ 0 n(xαδ(x 0 = n odd 0 n even, / = which gives us ψ ( = n odd,n> 4mα ( πnx π h n cos. (

2 P P0 0.5 m_lph hbr pi Figure : First order computtion of the probbility of finding the prticle t x = 0. (d (4 points To first order in α, the probbility of the prticles in the ground stte being found t x = 0 is given by ψ (0 = ψ (0 (0 + Re[ψ ( (0ψ(0 (0]. From prt (c, this is ψ (0 = ( + 4mα π h n. We need to evlute the infinite sum. To do this, it is useful to write n = so tht our sum is Thus, to first order, n odd,n> n = n odd,n> ψ (0 = ( n = n +. ( mα π h. ( n n+ We plot in figure the rtio ψ (0 / ψ (0 (0. In the region where mα π h, our pproximtion mkes good physicl sense: For α > 0, the potentil αδ(x is repulsive, nd so the probbility of finding the prticle t x = 0 decreses. For α < 0, the potentil is ttrctive, nd so the probbility of finding the prticle t x = 0 increses. However, t the point mα = π h /, vlue of α/ is equl to the ground stte energy, hence αδ(x cnnot be considered s perturbtion ny more. Notice tht beyond these vlues our first order pproximtion begins producing negtive probbilities, which is not trustworthy. (e (3 points The second order correction to the energies is gin 0 for even energy levels. For odd energy levels, we hve ( α E n ( m = π h n k. k odd,k n,

3 The second order correction to the ground stte energy in prticulr is ( E ( α m = π h k = m (mα π h. k odd,k> The second order correction to the ground stte energy is equl to the first order correction when (mα m π h = m (mα, tht is, when π h = mα. This estimte of the brekdown of perturbtion theory grees, up to fctors of O(, with our estimte of prt (d. (f (6 points We now wnt to solve the problem exctly. The boundry conditions on our wve function re: ( ψ(± = 0, (b ψ is continuous t x = 0, (c ψ x (ɛ ψ x mα ( ɛ = ψ(0. h We use the tril wve function { ψ(x = A e ikx + B e ikx / x 0 A e ikx + B e ikx 0 x /, where k = me/ h. Applying the first of our three conditions tells us, A + B e ik = A e ik + B = 0. ( Applying the second condition gives, A + B = A + B, using which with ( gives (B A ( e ik = 0. Therefore either exp(ik = or B = A. Now using the third boundry condition, we obtin: (A B (A B = mα ik h (A + B. (3 If exp(ik = then using ( we obtin A = B nd A = B, which substituted into (3 gives A = A. Hence, in this cse we deduce our wvefunctions re sin(k n x, where n is even nd k n = nπ/. This cse is sme s the unperturbed solution with even n. Hence there re no chnges either to the energy or to the wvefunctions with n even due to the delt function perturbtion. If B = A A, then ( gives A = Ae ik nd B = Ae ik. Substituting for A, B, A nd B in (3 we get trncendentl eqution: ( k cot = mα k h. (4 3

4 This implicitly determines the llowed vlues of k n, nd therefore the energies E n. The exct wvefunction is given by (not required for credit: { ψ cos(kn x mα n(x = k n h sin(k n x / x 0 cos(k n x + mα. k n h sin(k n x 0 x / We would now like to expnd (4 for smll α. This mens we need to know the expnsion of cot (x round x 0, cot (x / x + x 3 /3 + O(x 5, where n is n odd integer. This expnsion is not unique, becuse the cotngent is multivlued; we hve to specify the integer n ppering in the first term of the expnsion. Therefore we cn mrk these eigensttes by n odd n. Using this expnsion, k n + mα h k n We need to solve this order by order. To zeroth order in α, k n (0. To first order in α, k n ( + mα h k n (0 Therefore, up to first order in α the energy is + mα nπ h. E n = h k n m = h π n m + α + O(α, which is precisely the expnsion in α tht we found using the first order perturbtion theory. Optionl prts: Expnding to the second order in α k n ( + mα h k ( n Therefore energy up to the second order in α is (5 + mα nπ h m α h 4 n 3 π 3. (6 E n = h kn m = h π n m + α mα n π h + O(α3, which is exctly the energy upto the second order in α. To find the first order corrections to the wvefunction, let us evlute ψ n (0 ψ, which re co-efficients of expnsion c n. For even n you cn check tht they re going to generte second order corrections only. For odd n > we get, c n = ψ (0 n ψ = πn sin( mα sin( nπ π h (7 + 4 mα (n h π. (8 ( mα π h 4nmα ( n π h sin ( nπ Note: When we write the series for ψ (, using these coefficients c n, wht we get is similr to but NOT the sme s wht we obtined using perturbtion theory, given in Eq. (. Tht mens tht, somewhere, I hve mde slip. Perhps one of you cn figure out where. 4

5 3. A delt-function interction between two bosons in n infinite squre well (5 points ( ( points The prticles re independent, so to find the two prticle wve functions we simply multiply the free wve functions nd symmetrize: ψ ground = ψ (x ψ (x = ( cos πx cos ψ first = [ψ (x ψ (x + ψ (x ψ (x ] = [ ( πx cos sin ( πx + sin ( πx ( πx ( πx ] cos. The energies of these wve functions re the sums of the free prticle energies, E ground = h π m E first = 5 h π m. (b (3 points The first order correction to the ground stte energy is E ( ground = V 0 / = V 0 ( = 3V 0. / dx dx ψ (x ψ (x δ(x x / / ( dx cos 4 πx The first order correction to the first excited stte energy is E ( first = V / 0 / ( / = V 0 = V 0. dx dx ψ (x ψ (x + ψ (x ψ (x δ(x x / ( dx cos πx ( πx sin 4. Anhrmonic oscilltor (5 points ( (3 points The ground stte wve function is even, while the perturbtion λx 3 is odd. Therefore the integrl dx ψ 0 λx 3 vnishes, nd the first order contribution to the ground stte energy is zero. To clculte the second-order shift, we need the expecttion vlue n ( We clculte ( = , so the only contributions to the shift in the ground stte energy will come from the sttes nd 3. We find, therefore, ( 3 E ( h 0 = λ k ( ( 3 [ h 9 = λ mω E 0 E k mω hω + ], hω k>0 5

6 V x m omeg 3 lmbd x Figure : The potentil λx 3. which gives E ( 0 = λ h 8m 3 ω 4. (b (3 points The first order correction to the ground stte wve function is ψ ( 0 = m m H 0 E 0 E m m. As in prt (, the only contributions re from m = 3 nd m = : ψ ( 0 = λ ( ( 3/ h 3 + hω mω 3 3. ( The ground stte wve function is then, to first order, ψ 0 = 0 λ h 3/ ( hω mω (c (3 points A sketch of the potentil is in figure. This potentil is unbounded from below, nd so there is no ground stte ny stte loclized ner x = 0 is unstble, s it will eventully tunnel through the brrier. Perturbtion theory cnnot see tunneling effects. It is good for exmining reltively smll, loclized chnges in the potentil, but cnnot del with cses like this, where we hve drsticlly chnged the symptotic behvior of the potentil. The bove qulittive rgument is enough for full credit. However, it is nice to see quntittively wht is going on. You lerned in 8.04 how to compute the probbility of tunneling through brrier. If you pply these methods to our problem, you will find tht the probbility for the prticle to tunnel from x = 0 to x = mω λ goes like exp( const/λ. This is nonzero, but we cnnot expnd it s Tylor series in λ. Thus, to ny finite order in perturbtion theory, we compute the probbility of tunneling to be zero. (d (6 points We now tke the perturbtion to be H = λx 4. To clculte the first order shift in the ground stte energy, we need ( = We find the 6

7 V x V x x x Figure 3: The potentil mω x + λx 4, with (left λ > 0, nd (right λ < 0. first order shift in the ground stte energy is ( E ( h 0 = 3λ. mω We sketch the behvior of the potentil for positive nd negtive λ in figure 3. If λ > 0, perturbtion theory is good pproximtion to the ground stte, which sees region of the potentil where λx 4 mω x. (Perturbtion theory will not work so well for higher sttes. The contribution to the ground stte energy t first order is positive, which mkes physicl sense we re confining the prticles with shrper potentil, which suggests tht their energy should increse. If, of the other hnd, λ < 0, perturbtion theory clerly fils. The chnge to the ground stte energy is negtive, so perturbtion theory does see tht the overll energy will be lowered. However, s in prt (c, perturbtion theory doesn t know bout the chnge in the symptotic behvior of the potentil, nd cnnot inform us bout the tunneling tht will result. In cses like this, where flipping the sign of the perturbtion prmeter lters the symptotic behvior of the perturbtion, strictly speking the rdius of convergence of perturbtion theory s series in λ is zero. Therefore, even when λ > 0, perturbtion theory is n symptotic expnsion: it cnnot cpture ll the physics, lthough it cn still give us good informtion bout the first few terms in the series. 5. Polrizbility of prticle on ring (5 points ( (3 points The unperturbed Hmiltonin H 0 = h hs (normlized eigenfunctions ψ n (φ = π e inφ, for ll integers n. The energy eigenvlues re E n = h n m. Since the energy m φ depends only on n, the energies E n nd E n re degenerte, so ll energy levels re doubly degenerte except n = 0, which is non-degenerte. (b (5 points We cn use the usul non-degenerte perturbtion theory to clculte the first order correction to the ground stte wve function. We need the mtrix elements n cos φ 0, where I hve introduced the nottion n = Now cos φ = (eiφ + e iφ, nd π e inφ. 7

8 e ±iφ n = n ±, so n cos φ 0 = (δ n, + δ n,. Thus, ψ ( 0 = m3 qɛ ( +. h The induced electric dipole moment is ψ 0 q cos φ ψ 0 = q ψ (0 0 some lgebr we find to be d induced = m4 q h ɛ. The polrizbility of the system is, therefore, P = m4 q h. cos φ ψ( 0, which fter (c (7 points Becuse there re three hydrogen toms, if we rotte one CH 3 group by π/3, then the overll system is left invrint. Therefore our perturbtion should hve 3-fold rottion symmetry. Since tht is the simplest wy to mke sure H (φ + π/3 = H (φ. The simplest Hermitin opertor we could thus write down would be H cos 3φ (the choice H sin 3φ only differs by choice of origin. The chnge in the ground stte energy due to the perturbtion H = b cos 3φ is to first order 0. To second order, we hve E ( 0 = k = m b 9 h. m h k b 4 (δ k,3 + δ k, 3 The chnge in the ground stte wve function is to first order ψ ( 0 = m b h k δ k,3 + δ k,3 k k = m b ( h In position spce, the ground stte wve function is ψ 0 = minimized where cos 3φ =, nd mximized where cos 3φ =. π ( m b 9 h cos 3φ, so ψ 0 is Thus there is higher probbility of finding the orienttion of the CH 3 group t φ = π/3, π, 5π/3, nd lower probbility of finding it t φ = 0, π/3, 4π/3. This is esy to understnd physiclly, if we look t the ethne molecule end-on (see figure 4. The molecule cn lower its energy by rotting in such wy s to minimize the electrosttic energy rising from the interction between the hydrogen toms in the two groups. 8

9 Figure 4: A crtoon of n ethne molecule in its most fvorble orienttion, seen end on. 9

Massachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6

Massachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6 Msschusetts Institute of Technology Quntum Mechnics I (8.) Spring 5 Solutions to Problem Set 6 By Kit Mtn. Prctice with delt functions ( points) The Dirc delt function my be defined s such tht () (b) 3

More information

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions

Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis

More information

Name Solutions to Test 3 November 8, 2017

Name Solutions to Test 3 November 8, 2017 Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier

More information

PHY4605 Introduction to Quantum Mechanics II Spring 2005 Final exam SOLUTIONS April 22, 2005

PHY4605 Introduction to Quantum Mechanics II Spring 2005 Final exam SOLUTIONS April 22, 2005 . Short Answer. PHY4605 Introduction to Quntum Mechnics II Spring 005 Finl exm SOLUTIONS April, 005 () Write the expression ψ ψ = s n explicit integrl eqution in three dimensions, ssuming tht ψ represents

More information

221B Lecture Notes WKB Method

221B Lecture Notes WKB Method Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using

More information

Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1

Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1 Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

221A Lecture Notes WKB Method

221A Lecture Notes WKB Method A Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e

More information

Quantum Physics I (8.04) Spring 2016 Assignment 8

Quantum Physics I (8.04) Spring 2016 Assignment 8 Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter

More information

Physics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15

Physics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15 Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril

More information

Physics 215 Quantum Mechanics 1 Assignment 2

Physics 215 Quantum Mechanics 1 Assignment 2 Physics 15 Quntum Mechnics 1 Assignment Logn A. Morrison Jnury, 16 Problem 1 Clculte p nd p on the Gussin wve pcket α whose wve function is x α = 1 ikx x 1/4 d 1 Solution Recll tht where ψx = x ψ. Additionlly,

More information

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent? 1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the

More information

Continuous Quantum Systems

Continuous Quantum Systems Chpter 8 Continuous Quntum Systems 8.1 The wvefunction So fr, we hve been tlking bout finite dimensionl Hilbert spces: if our system hs k qubits, then our Hilbert spce hs n dimensions, nd is equivlent

More information

Physics Graduate Prelim exam

Physics Graduate Prelim exam Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is

More information

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1 Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,

More information

Quantum Physics II (8.05) Fall 2013 Assignment 2

Quantum Physics II (8.05) Fall 2013 Assignment 2 Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.

More information

The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr

The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues

More information

Physics 137A - Quantum Mechanics - Spring 2018 Midterm 1. Mathematical Formulas

Physics 137A - Quantum Mechanics - Spring 2018 Midterm 1. Mathematical Formulas Copyright c 8 by Austin J. Hedemn Physics 7A - Quntum Mechnics - Spring 8 Midterm Mondy, Februry 6, 6:-8: PM You hve two hours, thirty minutes for the exm. All nswers should be written in blue book. You

More information

Practice Problems Solution

Practice Problems Solution Prctice Problems Solution Problem Consier D Simple Hrmonic Oscilltor escribe by the Hmiltonin Ĥ ˆp m + mwˆx Recll the rte of chnge of the expecttion of quntum mechnicl opertor t A ī A, H] + h A t. Let

More information

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

More information

( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx"#t ( ) = 1 2" ( )"( x,t) PC 3101 Quantum Mechanics Section 1

( ) 2. ( ) is the Fourier transform of! ( x). ( ) ( ) ( ) = Ae i kx#t ( ) = 1 2 ( )( x,t) PC 3101 Quantum Mechanics Section 1 1. 1D Schrödinger Eqution G chpters 3-4. 1.1 the Free Prticle V 0 "( x,t) i = 2 t 2m x,t = Ae i kxt "( x,t) x 2 where = k 2 2m. Normliztion must hppen: 2 x,t = 1 Here, however: " A 2 dx " " As this integrl

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,

More information

dx x x = 1 and + dx α x x α x = + dx α ˆx x x α = α ˆx α as required, in the last equality we used completeness relation +

dx x x = 1 and + dx α x x α x = + dx α ˆx x x α = α ˆx α as required, in the last equality we used completeness relation + Physics 5 Assignment #5 Solutions Due My 5, 009. -Dim Wvefunctions Wvefunctions ψ α nd φp p α re the wvefunctions of some stte α in position-spce nd momentum-spce, or position representtion nd momentum

More information

Homework Problem Set 1 Solutions

Homework Problem Set 1 Solutions Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:

More information

Aike ikx Bike ikx. = 2k. solving for. A = k iκ

Aike ikx Bike ikx. = 2k. solving for. A = k iκ LULEÅ UNIVERSITY OF TECHNOLOGY Division of Physics Solution to written exm in Quntum Physics F0047T Exmintion dte: 06-03-5 The solutions re just suggestions. They my contin severl lterntive routes.. Sme/similr

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Deprtment 8.044 Sttisticl Physics I Spring Term 03 Problem : Doping Semiconductor Solutions to Problem Set # ) Mentlly integrte the function p(x) given in

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2011

Physics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2011 Physics 74 Grdute Quntum Mechnics Solutions to Finl Exm, Fll 0 You my use () clss notes, () former homeworks nd solutions (vilble online), (3) online routines, such s Clebsch, provided by me, or (4) ny

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

The graphs of Rational Functions

The graphs of Rational Functions Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior

More information

lim P(t a,b) = Differentiate (1) and use the definition of the probability current, j = i (

lim P(t a,b) = Differentiate (1) and use the definition of the probability current, j = i ( PHYS851 Quntum Mechnics I, Fll 2009 HOMEWORK ASSIGNMENT 7 1. The continuity eqution: The probbility tht prticle of mss m lies on the intervl [,b] t time t is Pt,b b x ψx,t 2 1 Differentite 1 n use the

More information

PH12b 2010 Solutions HW#3

PH12b 2010 Solutions HW#3 PH 00 Solutions HW#3. The Hmiltonin of this two level system is where E g < E e The experimentlist sis is H E g jgi hgj + E e jei hej j+i p (jgi + jei) j i p (jgi jei) ) At t 0 the stte is j (0)i j+i,

More information

1 1D heat and wave equations on a finite interval

1 1D heat and wave equations on a finite interval 1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion

More information

Chapter 3 The Schrödinger Equation and a Particle in a Box

Chapter 3 The Schrödinger Equation and a Particle in a Box Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics

More information

Problem Set 2 Solutions

Problem Set 2 Solutions Chemistry 362 Dr. Jen M. Stnr Problem Set 2 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt.).) opertor: /x ; function: x e

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

dt. However, we might also be curious about dy

dt. However, we might also be curious about dy Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

HW9.nb 1. be a symmetric (parity-even) function, with a close excited state with an anti-symmetric (parity-odd) wave function.

HW9.nb 1. be a symmetric (parity-even) function, with a close excited state with an anti-symmetric (parity-odd) wave function. HW9.nb HW #9. Rectngulr Double-Well Potentil Becuse the potentil is infinite for» x» > + b, the wve function should vnish t x = H + bl. For < x < + b nd - - b < x < -, the potentil vnishes nd the wve function

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

Recitation 3: More Applications of the Derivative

Recitation 3: More Applications of the Derivative Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Waveguide Guide: A and V. Ross L. Spencer

Waveguide Guide: A and V. Ross L. Spencer Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Chapter 28. Fourier Series An Eigenvalue Problem.

Chapter 28. Fourier Series An Eigenvalue Problem. Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Summary: Method of Separation of Variables

Summary: Method of Separation of Variables Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

More information

14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes

14.4. Lengths of curves and surfaces of revolution. Introduction. Prerequisites. Learning Outcomes Lengths of curves nd surfces of revolution 4.4 Introduction Integrtion cn be used to find the length of curve nd the re of the surfce generted when curve is rotted round n xis. In this section we stte

More information

Physics 220. Exam #1. April 21, 2017

Physics 220. Exam #1. April 21, 2017 Physics Exm # April, 7 Nme Plese red nd follow these instructions crefully: Red ll problems crefully before ttempting to solve them. Your work must be legible, nd the orgniztion cler. You must show ll

More information

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s

A5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s 4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

Abstract inner product spaces

Abstract inner product spaces WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

ragsdale (zdr82) HW2 ditmire (58335) 1

ragsdale (zdr82) HW2 ditmire (58335) 1 rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

IMPORTANT. Read these directions carefully:

IMPORTANT. Read these directions carefully: Physics 208: Electricity nd Mgnetism Finl Exm, Secs. 506 510. 7 My. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese

More information

Waveguides Free Space. Modal Excitation. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware

Waveguides Free Space. Modal Excitation. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware Modl Excittion Dniel S. Weile Deprtment of Electricl nd Computer Engineering University of Delwre ELEG 648 Modl Excittion in Crtesin Coordintes Outline 1 Aperture Excittion Current Excittion Outline 1

More information

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015

Prof. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015 Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be

More information

Week 10: Line Integrals

Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

Chapter 9 Many Electron Atoms

Chapter 9 Many Electron Atoms Chem 356: Introductory Quntum Mechnics Chpter 9 Mny Electron Atoms... 11 MnyElectron Atoms... 11 A: HrtreeFock: Minimize the Energy of Single Slter Determinnt.... 16 HrtreeFock Itertion Scheme... 17 Chpter

More information

Electromagnetism Answers to Problem Set 10 Spring 2006

Electromagnetism Answers to Problem Set 10 Spring 2006 Electromgnetism 76 Answers to Problem Set 1 Spring 6 1. Jckson Prob. 5.15: Shielded Bifilr Circuit: Two wires crrying oppositely directed currents re surrounded by cylindricl shell of inner rdius, outer

More information

Describe in words how you interpret this quantity. Precisely what information do you get from x?

Describe in words how you interpret this quantity. Precisely what information do you get from x? WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking out the wve function In quntum mechnics, the term wve function usully refers to solution to the Schrödinger eqution, Ψ(x, t) i = 2 2 Ψ(x, t) + V (x)ψ(x, t),

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

More information

Solutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7

Solutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7 Solutions to Problems in Merzbcher, Quntum Mechnics, Third Edition Homer Reid April 5, 200 Chpter 7 Before strting on these problems I found it useful to review how the WKB pproimtion works in the first

More information

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011 Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Chapter 5 : Continuous Random Variables

Chapter 5 : Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll

More information

NOTES ON HILBERT SPACE

NOTES ON HILBERT SPACE NOTES ON HILBERT SPACE 1 DEFINITION: by Prof C-I Tn Deprtment of Physics Brown University A Hilbert spce is n inner product spce which, s metric spce, is complete We will not present n exhustive mthemticl

More information

Ph2b Quiz - 1. Instructions

Ph2b Quiz - 1. Instructions Ph2b Winter 217-18 Quiz - 1 Due Dte: Mondy, Jn 29, 218 t 4pm Ph2b Quiz - 1 Instructions 1. Your solutions re due by Mondy, Jnury 29th, 218 t 4pm in the quiz box outside 21 E. Bridge. 2. Lte quizzes will

More information

) 4n+2 sin[(4n + 2)φ] n=0. a n ρ n sin(nφ + α n ) + b n ρ n sin(nφ + β n ) n=1. n=1. [A k ρ k cos(kφ) + B k ρ k sin(kφ)] (1) 2 + k=1

) 4n+2 sin[(4n + 2)φ] n=0. a n ρ n sin(nφ + α n ) + b n ρ n sin(nφ + β n ) n=1. n=1. [A k ρ k cos(kφ) + B k ρ k sin(kφ)] (1) 2 + k=1 Physics 505 Fll 2007 Homework Assignment #3 Solutions Textbook problems: Ch. 2: 2.4, 2.5, 2.22, 2.23 2.4 A vrint of the preceeding two-dimensionl problem is long hollow conducting cylinder of rdius b tht

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

Module 6: LINEAR TRANSFORMATIONS

Module 6: LINEAR TRANSFORMATIONS Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for

More information

Physics 202, Lecture 14

Physics 202, Lecture 14 Physics 202, Lecture 14 Tody s Topics Sources of the Mgnetic Field (Ch. 28) Biot-Svrt Lw Ampere s Lw Mgnetism in Mtter Mxwell s Equtions Homework #7: due Tues 3/11 t 11 PM (4th problem optionl) Mgnetic

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

We know that if f is a continuous nonnegative function on the interval [a, b], then b

We know that if f is a continuous nonnegative function on the interval [a, b], then b 1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

More information

Convergence of Fourier Series and Fejer s Theorem. Lee Ricketson

Convergence of Fourier Series and Fejer s Theorem. Lee Ricketson Convergence of Fourier Series nd Fejer s Theorem Lee Ricketson My, 006 Abstrct This pper will ddress the Fourier Series of functions with rbitrry period. We will derive forms of the Dirichlet nd Fejer

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

p(t) dt + i 1 re it ireit dt =

p(t) dt + i 1 re it ireit dt = Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)

More information

Recitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications

Recitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 Higher-Order Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht

More information

The Dirac distribution

The Dirac distribution A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution

More information

Reversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b

Reversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite

More information