Homework Problem Set 1 Solutions
|
|
- Cleopatra Skinner
- 5 years ago
- Views:
Transcription
1 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: x e x ( x ) e x = e x x e x b.) opertor: ; function: 4x + 9 The result is 4x no further simplifiction cn be one. c.) opertor: / ; function: x sin 3x ( x sin 3x) = ( sin 3x + 3x cos 3x) = 3cos 3x + 3cos 3x 9x sin 3x = 6 cos 3x 9x sin 3x.
2 . Consier two opertors, ˆ A = x n ˆ B = /. For the function f (x) = x e x, where is constnt, clculte the following quntities..) A ˆ B ˆ f ( x) A ˆ B ˆ f ( x) = x A ˆ B ˆ f x ( x e x ) ( ) = x e x x e x ( ) = x e x x e x. b.) B ˆ A ˆ f ( x) B ˆ A ˆ f ( x) = B ˆ A ˆ f x = x ( x ) e x ( x e x ) ( ) = x e x x e x. c.) Are the results from prts () n (b) the sme? If the results re not the sme, it is si tht the two opertors o not commute. No, in this cse, the results re not the sme. Tht is, A ˆ B ˆ f x opertors A ˆ n B ˆ o not commute. ( ) ˆ B A ˆ f x ( ). Therefore, the
3 3 3. The trnsltion opertor, ˆ T h, is efine by the rule: istnce of trnsltion..) Show whether or not ˆ T h is liner opertor. ˆ T h f (x) = f (x + h), where h represents the For linerity, we wnt to consier the ction of T ˆ h on the function given by f ( x) + b g( x) : ˆ T h Therefore the opertor ˆ T h is liner. [ f ( x) + b g( x) ] = f x + h ( ) + b g( x + h) = T ˆ h f ( x) + b T ˆ h g( x). b.) Evlute the expression T ˆ ( 1 3 T ˆ 1 + ) x, where h = 1, f (x) = x, n T ˆ h mens to pply the trnsltion opertor twice. ˆ T h f (x) = f (x + h), for f (x) = x n h = 1 we hve: T ˆ 1 f ( x) = f ( x +1), so T ˆ 1 x = ( x +1). Therefore, T ˆ ( 3 T ˆ ) x = T ˆ 1 x 3 ˆ T 1 x + x = T ˆ 1 ( x +1) 3 ( x +1) + x = ( x + ) 3 ( x +1) + x. The result cn be left in the form shown bove, or it cn be simplifie s shown below: T ˆ ( 3 T ˆ ) x = x + ( ) 3 ( x +1) + x ( ) + x = x + 4x x + x +1 = x +1.
4 4 4. Which of the following functions re eigenfunctions of /? e ikx, cos kx, kx, e x. In these functions, k n re constnts. For those functions tht re eigenfunctions, give the eigenvlues. () eikx = ike ikx The function e ikx is n eigenfunction of / with eigenvlue ik. (b) cos kx = k sin kx The function cos kx is not n eigenfunction of /. (c) kx = k The function kx is not n eigenfunction of /. () e x = x e x The function e x is not n eigenfunction of /.
5 5 5. Which of the following functions re eigenfunctions of eigenfunction.? Give the eigenvlues for ech.) e x ex = e x, the function e x is n eigenfunction of the opertor with eigenvlue 1. b.) x x =, the function x is NOT n eigenfunction of the opertor. c.) e x ( is constnt) The first erivtive is e x = x e x. The secon erivtive is e x = ( 4 x )e x. this result oes not equl constnt times the originl function, f (x) = e x is NOT n eigenfunction of the opertor /..) 3 cos x 3cos x ( ) = 3cos x = 1 3cos x ( ), the function 3 cos x is n eigenfunction of the opertor with eigenvlue 1. e.) sin x + cos x opertor ( sin x + cos x) = ( sin x + cos x), the function sin x + cos x is n eigenfunction of the with eigenvlue 1.
6 6 6. For prticle moving in three imensions, the clssicl expression for the kinetic energy T is given by: T = 1 m p x + p ( y + p z ). Give the corresponing expression for the quntum mechnicl kinetic energy opertor, ˆ T, in three imensions. To convert clssicl component of momentum, p α, where α = x, y,or z, into quntum mechnicl opertor ˆ p α, we use the reltion: p ˆ α = i!. α For the momentum squre (s ppers in the kinetic energy expression), we therefore hve: p ˆ α =! α. Substituting these momentum opertors, the quntum mechnicl kinetic energy opertor ˆ T becomes: ˆ T = 1 # m!!! & % $ y z (, ' n fctoring out the! terms gives: T ˆ =! m # % $ + y + & z (. ' ˆ T involves three vribles rther thn just one, it is more pproprite to use prtil erivtives: T ˆ =! m $ x + y + ' & % z ). ( 7. Inicte which functions correspon to cceptble quntum mechnicl wvefunctions on the intervl, [ ]..) x - not cceptble (goes to s x ) b.) e x - not cceptble (goes to s x ) c.) ex - not cceptble (goes to s x ).) e x ( is constnt) - cceptble
7 7 $ 8. Consier the wvefunction ψ(x) = sin nπ x ' & ), where n is positive integer n is constnt. % (.) Normlize the function ψ x ( ) on the intervl 0 to. [Note: you probbly will hve to look up the integrl in tble.] For normlize wvefunction Ψ norm, the integrl over ll spce must be equl to 1. Let Ψ norm = NΨ, where N is the normliztion constnt. For this problem, the rnge of the coorinte x is 0 to. Therefore, for normliztion we require: (x) = 1. Ψ norm 0 Substituting Ψ norm = NΨ les to N Ψ (x) = 1. 0 $ Substituting Ψ(x) = sin nπ x ' & ), we hve N sin # nπx & % ( = 1. To evlute the integrl, we cn look % ( 0 $ ' it up in tble. From the hnout of useful integrls, sin bx = x sinbx 4b. By mking the substitution b = nπ, we obtin: sin $ nπx ' & ) = 0 % ( + x $ nπx '. - sin& ) 0, 4nπ % (/ 0. Evluting t the limits yiels: sin $ nπx ' & ) = 0 % ( $ ' $ & sin( nπ )) & 0 % 4nπ ( % 4nπ sin ( 0 ) ' ). ( Note tht since sinnπ = 0 n sin0 = 0, the first n lst terms rop out, n we re left with sin $ nπx ' & ) = 0 % (. Finlly, substituting this result into the normliztion conition, N sin # nπx & % ( = 1, we hve 0 $ ' N " % $ ' = 1, or N = # &. Therefore, the normlize wvefunction Ψ norm is given by: Ψ norm = $ nπx ' sin& ). % (
8 8 8.) continue b.) Is ψ ( x) n eigenfunction of the momentum opertor, p ˆ x? p ˆ x = i!, the eigenvlue eqution woul be: ˆp x Ψ(x) = i! Ψ(x) = i! nπ x sin ˆp x Ψ(x) = i! nπ cos nπ x. the right sie is not equl to constnt times the originl function, Ψ(x) is NOT n eigenfunction of the opertor p ˆ x.
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
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
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
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,
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,
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
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
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
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
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
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
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
Notes on the Eigenfunction Method for solving differential equations
Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: Wereconsieringtheinfinite-imensionlHilbertspceL 2 ([, b] of ll squre-integrble functions over the intervl [, b] (ie, b f(x 2
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
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,
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
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
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
( ) 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
Basic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
Sturm-Liouville Theory
LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
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
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
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
x ) dx dx x sec x over the interval (, ).
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
Math 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
Chapter Five - Eigenvalues, Eigenfunctions, and All That
Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl
MATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion
Orthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
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
5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
Green function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
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,
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
If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as
Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x
When a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
ax bx c (2) x a x a x a 1! 2!! gives a useful way of approximating a function near to some specific point x a, giving a power-series expansion in x
Elementr mthemticl epressions Qurtic equtions b b b The solutions to the generl qurtic eqution re (1) b c () b b 4c (3) Tlor n Mclurin series (power-series epnsion) The Tlor series n n f f f n 1!! n! f
Quantum Physics III (8.06) Spring 2005 Solution Set 5
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
13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
Section 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
VII. The Integral. 50. Area under a Graph. y = f(x)
VII. The Integrl In this chpter we efine the integrl of function on some intervl [, b]. The most common interprettion of the integrl is in terms of the re uner the grph of the given function, so tht is
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
AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h
AM Mthemticl Anlysis Oct. Feb. Dte: October Exercises Lecture Exercise.. If h, prove the following identities hold for ll x: sin(x + h) sin x h cos(x + h) cos x h = sin γ γ = sin γ γ cos(x + γ) (.) sin(x
HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016
HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M
Math Fall 2006 Sample problems for the final exam: Solutions
Mth 42-5 Fll 26 Smple problems for the finl exm: Solutions Any problem my be ltered or replced by different one! Some possibly useful informtion Prsevl s equlity for the complex form of the Fourier series
Math 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5
Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to
The Fundamental Theorem of Calculus Part 2, The Evaluation Part
AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt
Math 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
NUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
4.5 THE FUNDAMENTAL THEOREM OF CALCULUS
4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere
CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS
CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics
Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
Bernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
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
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
Matrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
INTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
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
MA 201: Partial Differential Equations Lecture - 12
Two dimensionl Lplce Eqution MA 201: Prtil Differentil Equtions Lecture - 12 The Lplce Eqution (the cnonicl elliptic eqution) Two dimensionl Lplce Eqution Two dimensionl Lplce Eqution 2 u = u xx + u yy
Section 6.3 The Fundamental Theorem, Part I
Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt
5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
MIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model:
1 2 MIXED MODELS (Sections 17.7 17.8) Exmple: Suppose tht in the fiber breking strength exmple, the four mchines used were the only ones of interest, but the interest ws over wide rnge of opertors, nd
7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!
7. Problem 7. We hve two semi-innite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being
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
Homework Assignment 5 Solution Set
Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil
Introduction to Complex Variables Class Notes Instructor: Louis Block
Introuction to omplex Vribles lss Notes Instructor: Louis Block Definition 1. (n remrk) We consier the complex plne consisting of ll z = (x, y) = x + iy, where x n y re rel. We write x = Rez (the rel prt
University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009
University of Wshington Deprtment of Chemistry Chemistry Winter Qurter 9 Homework Assignment ; Due t pm on //9 6., 6., 6., 8., 8. 6. The wve function in question is: ψ u cu ( ψs ψsb * cu ( ψs ψsb cu (
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.
Sections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
Conservation Law. Chapter Goal. 6.2 Theory
Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the
SPECIALIST MATHEMATICS
Victorin Certificte of Euction 08 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written exmintion Tuesy 5 June 08 Reing time:.00 pm to.5 pm (5 minutes) Writing
u t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
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
Instantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
Best Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
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
1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang
nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution
x dx does exist, what does the answer look like? What does the answer to
Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl
September 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
Separation of Variables in Linear PDE
Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the one-dimensionl (1D) wve eqution 2 u
M344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
Mass Creation from Extra Dimensions
Journl of oern Physics, 04, 5, 477-48 Publishe Online April 04 in SciRes. http://www.scirp.org/journl/jmp http://x.oi.org/0.436/jmp.04.56058 ss Cretion from Extr Dimensions Do Vong Duc, Nguyen ong Gio
Course 2BA1 Supplement concerning Integration by Parts
Course 2BA1 Supplement concerning Integrtion by Prts Dvi R. Wilkins Copyright c Dvi R. Wilkins 22 3 The Rule for Integrtion by Prts Let u n v be continuously ifferentible rel-vlue functions on the intervl
Section 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
Linear Differential Equations Physics 129a Solutions to Problems Frank Porter Revision F. Porter
Liner Differentil Equtions Physics 19 Solutions to Problems 051018 Frnk Porter Revision 11106 F. Porter 1 Exercises 1. Consider the generl liner second order homogeneous differentil eqution in one dimemsion:
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
NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
Definite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
Math 142: Final Exam Formulas to Know
Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be
1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
Introduction to Some Convergence theorems
Lecture Introduction to Some Convergence theorems Fridy 4, 005 Lecturer: Nti Linil Notes: Mukund Nrsimhn nd Chris Ré. Recp Recll tht for f : T C, we hd defined ˆf(r) = π T f(t)e irt dt nd we were trying
Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function