Chapter Seven The Quantum Mechanical Simple Harmonic Oscillator
|
|
- Shanon Wade
- 6 years ago
- Views:
Transcription
1 Capter Seven Te Quantum Mecanical Simple Harmonic Oscillator Introduction Te potential energy function for a classical, simple armonic oscillator is given by ZÐBÑ œ 5B were 5 is te spring constant. Suc a classical oscillator as an angular frequency = œ È5Î7, were 7 is te mass of te oscillator. Writing te potential energy function in terms of te angular frequency, rater tan te spring constant, gives ZÐBÑ œ B (7.1) Te time-independent Scrodinger equation can ten be written in te form or, in operator notation, ` < ÐBÑ B < ÐBÑ œ I< ÐBÑ (7.2) 7 `B s: < ÐBÑ sb < ÐBÑ œ I< ÐBÑ (7.3) 7 Te potential energy function for tis problem is different from te ones we ave considered tus far. Tis potential energy function essentially spans all space. In te diagram below, you can see tat tere are regions were te total energy is greater tan te potential (regions were te wave function would be sinusoidal) and regions were te total energy is less tan te potential (regions were proper wave functions will be exponentially decreasing). Tus, a proper solution to te simple armonic oscillator problem will be sinusoidal in te central region, but must be exponentially decreasing as B tends toward bot positive and negative infinity. Fig. 7.1 Te Simple Harmonic Oscillator Potential
2 Quantum Harmonic Oscillator 2 Analytical Metod We begin wit te time-independent Scrodinger equation for te simple armonic oscillator, expressed in terms of te angular frequency ` < ÐBÑ B < ÐBÑ œ I< ÐBÑ (7.) 7 `B In order to simplify tis equation somewat, we will let B œ α (7.5) Tis introduces an arbitrary constant α wic can ten be defined is suc a way as to create a dimensionless equation. Te Scrodinger equation becomes ` Ð Ñ 7 Ð Ñ œ I Ð Ñ 7 α ` < = α < < (7.6) Multiplying by 7α Î gives % ` 7 7 ` Ð Ñ < = α < Ð Ñ œ α I< ÐÑ (7.7) ow using our freedom to define α, we coose α to be so tat our equation simplifies to te dimensionless form α œ È Î (7.8) ` < ÐÑ < Ð Ñ œ I< Ð Ñ (7.9) ` = You can tell tis equation is indeed dimensionless, since te energy as units of =. Letting O œ IÎ= our equation reduces to ote tat ` Ð Ñ œ O Ð Ñ ` < ˆ < (7.1) For large (i.e., large displacements from te equilibrium position were p ) we expect te wave function to ave te approximate form < ÐÑ / (7.11) since we know tat te wave function must tend toward zero for bot positive and negative values of. Let's take tis approximate form of te wave function as a trial
3 Quantum Harmonic Oscillator 3 solution to our differential equation and see ow well it satisfies our differential equation. ` ` / œ / ` ` œ / ˆ % / œ % / Œ (7.12) otice tat if we set œ ß tis last equation reduces to ` / œ ˆ / (7.13) ` wic is almost exactly te form of our dimensionless oscillator equation except for te energy constant O. We expect te exact solution of our differential equation, terefore, to be of te form < ÐÑ œ 2ÐÑ/ (7.1) and ope tat pulling out te exponential term will elp us find a simple expression for 2Ð Ñ. (One can always ope) Our dimensionless diffential equation for te quantum oscillator is ` Ð Ñ œ O Ð Ñ ` < ˆ < (7.15) If we assum tat te correct solution for tis equation as te form of equation 7.1, we need te evaluate te second partial of tis function. Te first partial is given by ` ` Š 2ÐÑ/ œ 2ÐÑ / Œ 2ÐÑŠ / ` ` `2ÐÑ œ Œ 2ÐÑ / ` (7.16) Taking te partial again, we obtain 2 ` < ÐÑ ` 2ÐÑ `2ÐÑ `2ÐÑ œ / / / Š Š ` ` ` ` 2ÐÑŠ / Š / 2 ` < ÐÑ ` 2ÐÑ `2ÐÑ œ / 2Ð ` œ ш ` ` (7.17) Our dimensionless differential equation for te quantum oscillator becomes 2 ` 2Ð Ñ `2Ð Ñ / 2Ð Ñ œ O 2Ð Ñ/ œ ˆ ` ˆ (7.18) `
4 Quantum Harmonic Oscillator Tis leads to a differential equation for te function out, given by 2ÐÑß since te exponential drops ` 2 2Ð Ñ `2Ð Ñ O 2ÐÑ œ (7.19) ` ` Tis equation we now solve using te assumption tat te solution can be expressed in terms of a power series in of te form 2ÐÑ œ â œ + (7.2) Differentiating te series term by term, onceß gives `2ÐÑ ` œ + + $+ â œ + (7.21) $ Differentiating again gives 2 ` 2ÐÑ œ + $ + $ % + â ` $ % œ + (7.22) Substituting tis into te differential equation for 2ÐÑ gives + + O + œ (7.23) Collecting terms of te same power and rearranging finally gives or + + O + œ (7.2) c + + O + d œ (7.25) ow if te differential equation must be valid for all values of te variable, ten te only way tis last equation can be valid for any arbitrary coice of is for te coefficients to be indentically zero, or + + O + œ (7.26)
5 Quantum Harmonic Oscillator 5 ow tis equation relates te coefficient + to te coefficient + according to te equation O + œ + (7.27) Tis recursion formula enables us to write any coefficient if we know just two, + and +. All oter coefficents can be determined based upon tese two. We terefore write were 2ÐÑ œ 2evenÐÑ2odd ÐÑ (7.28) % 2evenÐÑ œ % â is te even function of, built upon +, and were (7.29) $ & 2oddÐÑ œ + + $ + & â (7.3) is te odd function of, built upon +. Tus, we ave te solution of te differential equation in terms of two undetermined constants, + and +, wic is wat we would expect from a second-order differential equation. Te final form of te solution, ten, is < ÐÑ œ / 2ÐÑ œ / + + Ÿ (7.31) were we use te recursion relation to determine te + 's from + and +. Before we celebrate too muc, owever, we must determine if tis wave function converges as p ; oterwise te solution we ave obtained is unacceptable. If tere is a problem wit tis function as gets larger it would obviously be due to te larger powers of, i.e., larger values of. Let's see wat te recursion formula gives us as gets large. Te recursion formula can be written O + œ + ÎOÎ œ + Î Î (7.32) or, in te limit of large lim + œ + large Ä (7.33) or lim Älarge + œ (7.3) + If we let be te value of were tis limit begins to be valid, ten we can divide te summation up into two separate pieces; te region below were te recursion formula
6 Quantum Harmonic Oscillator 6 must be used, and te region above were te approximation to te recursion formula for large can be used. Te summation over te even terms, ten, becomes œ 2evenÐÑ œ + œ + + (7.35) or % ' + œ â % ' But for large we ave + % œ + ' œ + + % â % + œ ' + % + % + % œ œ % % (7.36) (7.37) giving % + œ + + œ â % + œ + + xœ â x x x % + œ + + xœ â x x x % + œ + + xœ â x x x + œ + + x â x x x Ÿ ÐÑ + œ + + x x x x â Ÿ + œ + + x š / ÐÑ (7.38)
7 Quantum Harmonic Oscillator 7 form Te even solution to te quantum armonic oscillator problem, terefore, as te < evenðñ œ / 2evenÐÑ œ / + + x š / Ÿ < ÐÑ œ / 2 ÐÑ œ / + G / even even (7.39) wic clearly diverges. Tis same type beavior can be demonstrated for < oddðñ as well. So wat do we do? Obviously, our solution is not te infinite series we ave derived, because tis infinite series diverges. However, if te series were to terminate for some value of, ten te function would remain finite. Let's look again at te recursion formula O + œ + (7.) If we demand tat te term + œ for œ - (were + Á Ñ, we obtain -O - - œ (7.1) or - O œ (7.2) from wic we obtain an equation wic fixes te energy of te oscillator or O œ IÎ œ - = - (7.3) I- œ Œ- = were - œ ßßß$ßá (7.) Tis gives us a denumerable energy spectrum and implies tat te correct solution for our problem is not an infinite series solution for 2ÐÑ, but a polynomial solution for 2Ð Ñ, expressed as 2 ÐÑ œ â œ + (7.5) - wit different order polynomials being associated wit different energies. ow te coefficients satisfy te recursion formula -
8 Quantum Harmonic Oscillator 8 O + œ + (7.6) were O œ -, so te recursion relation depends upon - as well as. We express tis recursion formula wit tis substitution to obtain - + œ (7.7) For - œ, Ðcorresponding to O œ, wic gives I œ = ), and we see tat te recursion formular + œ + (7.8) gives zero for all even + 's oter tan +. But wat about te odd values of +? We require tat + be set equal to zeroà oterwise, we would ave an infinite series wic diverges. Likewise, + is set equal to zero for te odd polynomials. Te polynomials, ten can be expressed as: 2 ÐÑ œ + 2 ÐÑ œ + 2 ÐÑ œ ÐÑ œ + + $ $ $ $ $ % % % % 2% ÐÑ œ % ã (7.9) or, in terms only of te first coefficient in eac of te even or odd sequences (note tat % + Á + Á + Ñ 2 ÐÑ œ + 2 ÐÑ œ + 2 ÐÑ œ + (7.5) ã $ $ 2 Ð Ñ œ + $ œ $ % % 2 Ð Ñ œ + % % % œ $ Remember tat multiplying te wavefunction solution to Scrödinger's equation by some arbitrary constant does not cange te differential equation. Tis is te reason we are always at liberty to normalize te wave function. Te leading coefficient, ten, (te one outside te braces) in eac of tese polynomials will be determined by normalization of te wave function. Tis means tat we are at liberty to multiply or divide te coefficients in eac polynomial by some common factor witout canging te
9 Quantum Harmonic Oscillator 9 equation in any significant way (we just incorporate tat arbitrary cange into te arbitrary coefficient). It is customary to coose te constants in te polynomials so tat te coefficient of te igest order term of is 8. Doing tis removes te fractions in te polynomial equations. In addition, we can mulitiply by a negative so tat te sign of te igest order term is always positive. Wit tese canges, te polynomials can be written as 2 ÐÑ œ + 2 ÐÑ œ + 2 ÐÑ œ + % $ $ 2$ ÐÑ œ + ) % % 2% ÐÑ œ + ' %) ã (7.51) Te terms inside te braces turn out to be te so-called Hermite polynomials wic arise from te Taylor series expansion 8 D D D / œ L8ÐÑ (7.52) 8x 8œ and can be generated using te Rodrigues formula 8 8. L8ÐÑ œ / Œ /. (7.53) Problem 7.1 Sow tat you can obtain te terms in te braces from te Taylor series expansion. Also, prove tat te Rodrigues formula works for te first few Hermite polynomials. So, apart from te normalization constant, te solutions to te armonic oscillator problem are just te Hermite polynomials. We next need to determine te normalization constants. Te wave functions associated wit te various energy states ave te form < ÐÑ œ 2 ÐÑ/ 8 8 (7.5) Te normalization condition for te ground state wave function is * * ( < ÐBÑ< ÐBÑ.B œ Ê ( < Ð Ñ< ÐÑ. œ (7.55)
10 Quantum Harmonic Oscillator 1 were we recall tat B œ α, wit α œ ÈÎ. Substituting in for te wave function, we ave Ê ( ˆ + /. œ Ê ˆ ( 7 + / =. œ Ê ˆ + œ 1 œ 7 = (7.56) using te Gaussian integral tables. Solving for te coefficient, we ave + œ 7 = 1 ormalization of te next wave function gives Ê ( Ê ( ˆ + /. œ Ê ˆ ( 7 + % / =. œ % * < Ð Ñ< ÐÑ. œ Ê ˆ + ) œ 1 œ 7 = % (7.57) (7.58) or, solving for te coefficient, 7 = + œ œ È 1 È x 1 % % (7.59) Continuing tis process, we find tat te normalization constant can be written E œ 8 7 È 88x = 1 % (7.6) Tus, te solution to te quantum armonic oscillator problem can be written < 8ÐÑ œ L 8/ È 88x 1 % Î (7.61) or since B œ α, wit α œ ÈÎ, we write œ È ÎB so tat te equation, in terms of B can be written < 8ÐBÑ œ L8 Î B / È 88x 1 ˆ È % B Î (7.62)
11 Quantum Harmonic Oscillator 11 ote 1: ormalization of te second order wave function gives Ê ( * < Ð Ñ< ÐÑ. œ Ê ( + % /. œ Ê ˆ + ( % ' ' % /. œ 7 = Ê ˆ $ + œ ' 1 ' 1 % 1 œ 7 = ) % 1 Ê ˆ 7 = + e '% f œ 1 Ê ˆ 7 = + ) œ or, solving for te coefficient, 7 = + œ œ È) 1 È x 1 % % (7.63) wic follows te form stated above. ote 2: Two additional relationsips for te Hermite polynomials wic often prove useful in calculations are L œ L 8L (7.6) were one polynomial can be related to te sum of two previously calculated polynomials, and `L8 œ 8L ` 8 (7.65) were te derivative of a polynomial is related to te previous polynomial.
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More informationPhysically Based Modeling: Principles and Practice Implicit Methods for Differential Equations
Pysically Based Modeling: Principles and Practice Implicit Metods for Differential Equations David Baraff Robotics Institute Carnegie Mellon University Please note: Tis document is 997 by David Baraff
More informationChapter 1D - Rational Expressions
- Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere
More information232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More informationPrecalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
More information4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
More informationNotes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
More informationLIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
More informationMath 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
More informationMATH1151 Calculus Test S1 v2a
MATH5 Calculus Test 8 S va January 8, 5 Tese solutions were written and typed up by Brendan Trin Please be etical wit tis resource It is for te use of MatSOC members, so do not repost it on oter forums
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More informationContinuity and Differentiability of the Trigonometric Functions
[Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te
More informationCopyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More information2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
More informationRecall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if
Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More information5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
More informationLogarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
More information1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
More information. If lim. x 2 x 1. f(x+h) f(x)
Review of Differential Calculus Wen te value of one variable y is uniquely determined by te value of anoter variable x, ten te relationsip between x and y is described by a function f tat assigns a value
More informationMathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative
Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x
More informationContinuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
More information2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
More information3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
More informationA h u h = f h. 4.1 The CoarseGrid SystemandtheResidual Equation
Capter Grid Transfer Remark. Contents of tis capter. Consider a grid wit grid size and te corresponding linear system of equations A u = f. Te summary given in Section 3. leads to te idea tat tere migt
More informationMAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
More informationMaterial for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More information4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
More informationSolve exponential equations in one variable using a variety of strategies. LEARN ABOUT the Math. What is the half-life of radon?
8.5 Solving Exponential Equations GOAL Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT te Mat All radioactive substances decrease in mass over time. Jamie works in
More informationAnalytic Functions. Differentiable Functions of a Complex Variable
Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general
More informationThe total error in numerical differentiation
AMS 147 Computational Metods and Applications Lecture 08 Copyrigt by Hongyun Wang, UCSC Recap: Loss of accuracy due to numerical cancellation A B 3, 3 ~10 16 In calculating te difference between A and
More informationSolution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
More informationLECTURE 14 NUMERICAL INTEGRATION. Find
LECTURE 14 NUMERCAL NTEGRATON Find b a fxdx or b a vx ux fx ydy dx Often integration is required. However te form of fx may be suc tat analytical integration would be very difficult or impossible. Use
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationPractice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
More informationRunge-Kutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t))
Runge-Kutta metods Wit orders of Taylor metods yet witout derivatives of f (t, y(t)) First order Taylor expansion in two variables Teorem: Suppose tat f (t, y) and all its partial derivatives are continuous
More informationExercises for numerical differentiation. Øyvind Ryan
Exercises for numerical differentiation Øyvind Ryan February 25, 2013 1. Mark eac of te following statements as true or false. a. Wen we use te approximation f (a) (f (a +) f (a))/ on a computer, we can
More informationPoisson Equation in Sobolev Spaces
Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on
More information2.3 Product and Quotient Rules
.3. PRODUCT AND QUOTIENT RULES 75.3 Product and Quotient Rules.3.1 Product rule Suppose tat f and g are two di erentiable functions. Ten ( g (x)) 0 = f 0 (x) g (x) + g 0 (x) See.3.5 on page 77 for a proof.
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare ttp://ocw.mit.edu 5.74 Introductory Quantum Mecanics II Spring 9 For information about citing tese materials or our Terms of Use, visit: ttp://ocw.mit.edu/terms. Andrei Tokmakoff, MIT
More informationlecture 26: Richardson extrapolation
43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)
More informationHow to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
More informationSection 3.1: Derivatives of Polynomials and Exponential Functions
Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it
More informationChapter 2 Limits and Continuity
4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(
More informationDerivatives of trigonometric functions
Derivatives of trigonometric functions 2 October 207 Introuction Toay we will ten iscuss te erivates of te si stanar trigonometric functions. Of tese, te most important are sine an cosine; te erivatives
More informationLecture 15. Interpolation II. 2 Piecewise polynomial interpolation Hermite splines
Lecture 5 Interpolation II Introduction In te previous lecture we focused primarily on polynomial interpolation of a set of n points. A difficulty we observed is tat wen n is large, our polynomial as to
More informationAMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.
Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error
More informationThe derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
More informationA.P. CALCULUS (AB) Outline Chapter 3 (Derivatives)
A.P. CALCULUS (AB) Outline Capter 3 (Derivatives) NAME Date Previously in Capter 2 we determined te slope of a tangent line to a curve at a point as te limit of te slopes of secant lines using tat point
More informationMTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
More informationExam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
More information158 Calculus and Structures
58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you
More information(4.2) -Richardson Extrapolation
(.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as
More informationDerivatives of Exponentials
mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function
More informationFinite Difference Methods Assignments
Finite Difference Metods Assignments Anders Söberg and Aay Saxena, Micael Tuné, and Maria Westermarck Revised: Jarmo Rantakokko June 6, 1999 Teknisk databeandling Assignment 1: A one-dimensional eat equation
More informationCalculus I Practice Exam 1A
Calculus I Practice Exam A Calculus I Practice Exam A Tis practice exam empasizes conceptual connections and understanding to a greater degree tan te exams tat are usually administered in introductory
More informationA Reconsideration of Matter Waves
A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,
More informationMath Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
More informationSFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00
SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only
More informationThe Laplace equation, cylindrically or spherically symmetric case
Numerisce Metoden II, 7 4, und Übungen, 7 5 Course Notes, Summer Term 7 Some material and exercises Te Laplace equation, cylindrically or sperically symmetric case Electric and gravitational potential,
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )
More information1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.
004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following
More information1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist
Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter
More informationMath 1241 Calculus Test 1
February 4, 2004 Name Te first nine problems count 6 points eac and te final seven count as marked. Tere are 120 points available on tis test. Multiple coice section. Circle te correct coice(s). You do
More informationRules of Differentiation
LECTURE 2 Rules of Differentiation At te en of Capter 2, we finally arrive at te following efinition of te erivative of a function f f x + f x x := x 0 oing so only after an extene iscussion as wat te
More informationNumerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1
Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.
More informationOrder of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
More informationMathematics 105 Calculus I. Exam 1. February 13, Solution Guide
Matematics 05 Calculus I Exam February, 009 Your Name: Solution Guide Tere are 6 total problems in tis exam. On eac problem, you must sow all your work, or oterwise torougly explain your conclusions. Tere
More information(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?
Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is
More informationREVIEW LAB ANSWER KEY
REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g
More information8œ! This theorem is justified by repeating the process developed for a Taylor polynomial an infinite number of times.
Taylor and Maclaurin Series We can use the same process we used to find a Taylor or Maclaurin polynomial to find a power series for a particular function as long as the function has infinitely many derivatives.
More information1 (10) 2 (10) 3 (10) 4 (10) 5 (10) 6 (10) Total (60)
First Name: OSU Number: Last Name: Signature: OKLAHOMA STATE UNIVERSITY Department of Matematics MATH 2144 (Calculus I) Instructor: Dr. Matias Sculze MIDTERM 1 September 17, 2008 Duration: 50 minutes No
More information5.1 introduction problem : Given a function f(x), find a polynomial approximation p n (x).
capter 5 : polynomial approximation and interpolation 5 introduction problem : Given a function f(x), find a polynomial approximation p n (x) Z b Z application : f(x)dx b p n(x)dx, a a one solution : Te
More informationSymmetry Labeling of Molecular Energies
Capter 7. Symmetry Labeling of Molecular Energies Notes: Most of te material presented in tis capter is taken from Bunker and Jensen 1998, Cap. 6, and Bunker and Jensen 2005, Cap. 7. 7.1 Hamiltonian Symmetry
More informationQuantum Mechanics Chapter 1.5: An illustration using measurements of particle spin.
I Introduction. Quantum Mecanics Capter.5: An illustration using measurements of particle spin. Quantum mecanics is a teory of pysics tat as been very successful in explaining and predicting many pysical
More informationChapter 2. Limits and Continuity 16( ) 16( 9) = = 001. Section 2.1 Rates of Change and Limits (pp ) Quick Review 2.1
Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 969) Quick Review..... f ( ) ( ) ( ) 0 ( ) f ( ) f ( ) sin π sin π 0 f ( ). < < < 6. < c c < < c 7. < < < < < 8. 9. 0. c < d d < c
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More informationWalrasian Equilibrium in an exchange economy
Microeconomic Teory -1- Walrasian equilibrium Walrasian Equilibrium in an ecange economy 1. Homotetic preferences 2 2. Walrasian equilibrium in an ecange economy 11 3. Te market value of attributes 18
More informationMath 161 (33) - Final exam
Name: Id #: Mat 161 (33) - Final exam Fall Quarter 2015 Wednesday December 9, 2015-10:30am to 12:30am Instructions: Prob. Points Score possible 1 25 2 25 3 25 4 25 TOTAL 75 (BEST 3) Read eac problem carefully.
More informationLIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
More information2.1 THE DEFINITION OF DERIVATIVE
2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative
More informationChapters 19 & 20 Heat and the First Law of Thermodynamics
Capters 19 & 20 Heat and te First Law of Termodynamics Te Zerot Law of Termodynamics Te First Law of Termodynamics Termal Processes Te Second Law of Termodynamics Heat Engines and te Carnot Cycle Refrigerators,
More informationApplications of the van Trees inequality to non-parametric estimation.
Brno-06, Lecture 2, 16.05.06 D/Stat/Brno-06/2.tex www.mast.queensu.ca/ blevit/ Applications of te van Trees inequality to non-parametric estimation. Regular non-parametric problems. As an example of suc
More informationDigital Filter Structures
Digital Filter Structures Te convolution sum description of an LTI discrete-time system can, in principle, be used to implement te system For an IIR finite-dimensional system tis approac is not practical
More informationALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU. A. Fundamental identities Throughout this section, a and b denotes arbitrary real numbers.
ALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU A. Fundamental identities Trougout tis section, a and b denotes arbitrary real numbers. i) Square of a sum: (a+b) =a +ab+b ii) Square of a difference: (a-b)
More informationCalculus I - Spring 2014
NAME: Calculus I - Spring 04 Midterm Exam I, Marc 5, 04 In all non-multiple coice problems you are required to sow all your work and provide te necessary explanations everywere to get full credit. In all
More informationSection 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.
Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope
More informationDifferential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
More information160 Chapter 3: Differentiation
3. Differentiation Rules 159 3. Differentiation Rules Tis section introuces a few rules tat allow us to ifferentiate a great variety of functions. By proving tese rules ere, we can ifferentiate functions
More informationAppendix 4.1 Alternate Solution to the Infinite Square Well with Symmetric Boundary Conditions. V(x) I II III +L/2
Appendix 4.1 Alternate Solution to the Infinite Square Well with Symmetric Boundary Conditions V(x) I II III -L/2 +L/2 x In this problem, a particle of mass 7 confined by a potential energy which has a
More informationIntegral Calculus, dealing with areas and volumes, and approximate areas under and between curves.
Calculus can be divided into two ke areas: Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and minima problems Integral
More information