DOCSLIB.ORG

Numerical Methods in Quantum Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numerical Methods in Quantum Mechanics Lecture notes Numerical Methods in Quantum Mechanics Corso di Laurea Magistrale in Fisica Interateneo Trieste { Udine Anno accademico 2020/2021 Paolo Giannozzi University of Udine Contains software and material written by Furio Ercolessi1 and Stefano de Gironcoli2 1Formerly at University of Udine 2SISSA - Trieste Last modified May 27, 2021 Contents Introduction 1 0.1 About Software . .1 0.1.1 Compilers . .1 0.1.2 Visualization Tools . .2 0.1.3 Mathematical Libraries . .2 0.1.4 Pitfalls in C-Fortran interlanguage calls . .3 0.2 Bibliography . .4 1 One-dimensional Schr¨odingerequation 5 1.1 The harmonic oscillator . .5 1.1.1 Units . .6 1.1.2 Exact solution . .6 1.1.3 Comparison with classical probability density . .8 1.2 Quantum mechanics and numerical codes . .9 1.2.1 Quantization . .9 1.2.2 A pitfall: pathological asymptotic behavior . .9 1.3 Numerov's method . 10 1.3.1 Code: harmonic0 . 12 1.3.2 Code: harmonic1 . 13 1.3.3 Laboratory . 15 2 Schr¨odingerequation for central potentials 16 2.1 Variable separation . 16 2.1.1 Radial equation . 18 2.2 Coulomb potential . 18 2.2.1 Energy levels . 20 2.2.2 Radial wave functions . 20 2.3 Code: hydrogen radial . 21 2.3.1 Logarithmic grid . 21 2.3.2 Improving convergence with perturbation theory . 22 2.3.3 Laboratory . 24 3 Scattering from a potential 25 3.1 Short reminder of the theory of scattering . 25 3.2 Scattering of H atoms from rare gases . 27 3.3 Code: crossection . 28 i 3.3.1 Laboratory . 30 4 The Variational Method 31 4.1 The variational principle . 31 4.1.1 Demonstration of the variational principle . 31 4.1.2 Alternative demonstration of the variational principle . 32 4.1.3 Variational principle for the ground state energy . 33 4.2 The variational method in practice . 34 4.2.1 Expansion into a basis set of orthonormal functions . 35 4.2.2 Secular equation . 36 4.3 Plane-wave basis set . 37 4.4 Code: pwell . 38 4.4.1 Diagonalization routines . 39 4.4.2 Laboratory . 40 5 Non-orthonormal basis sets 41 5.1 Variational method for non-orthonormal basis set . 41 5.1.1 Gaussian basis set . 43 5.1.2 Other kinds of localized basis functions . 43 5.2 Code: hydrogen gauss . 44 5.2.1 Laboratory . 45 6 Self-consistent field 47 6.1 The Hartree-Fock method . 47 6.1.1 Slater determinants . 48 6.1.2 Hartree-Fock equations . 48 6.1.3 Hartree and exchange potentials . 50 6.1.4 Hartree-Fock and correlation energy . 51 6.1.5 Helium atom . 52 6.2 Code: helium hf radial . 53 6.2.1 Laboratory . 54 7 Molecules 55 7.1 Born-Oppenheimer approximation . 55 7.2 Potential Energy Surface . 56 7.3 Diatomic molecules . 57 7.4 Rothaan-Hartree-Fock equations . 58 7.4.1 Multi-center Gaussian integrals . 60 7.5 Code: h2 hf gauss . 61 7.5.1 Laboratory . 62 8 Electrons in periodic potential 63 8.1 Crystalline solids . 63 8.1.1 Periodic Boundary Conditions . 64 8.1.2 Bloch Theorem . 65 8.1.3 The empty potential . 65 8.1.4 Solution for the crystal potential . 66 ii 8.1.5 Plane-wave basis set . 67 8.2 Code: periodicwell . 69 8.2.1 Laboratory . 70 9 Pseudopotentials 72 9.1 Three-dimensional crystals . 72 9.2 Plane waves, core states, pseudopotentials . 73 9.3 Code: cohenbergstresser . 74 9.3.1 Laboratory . 76 10 Exact diagonalization of quantum spin models 77 10.1 The Heisenberg model . 77 10.2 Hilbert space in spin systems . 78 10.3 Iterative diagonalization . 79 10.4 Code: heisenberg exact ...................... 80 10.4.1 Computer Laboratory . 82 11 Density-Functional Theory 83 11.1 Hohenberg-Kohn theorem . 83 11.2 Kohn-Sham equations . 84 11.3 Approximated functionals . 85 11.4 Structure of a DFT code . 86 11.4.1 Matrix elements of the potential . 87 11.4.2 FFT and FFT grids . 88 11.4.3 Computing the charge density . 89 11.4.4 Computing the potential . 90 11.4.5 Laboratory . 90 A Real-space two- and three-dimensional grids 92 B Solution of time-dependent Schr¨odingerequations 94 B.1 Discretization in time: Crank-Nicolson algorithm . 95 B.2 Direct discretization of the time evolution operator . 96 C Derivation of Van der Waals interaction 98 D The Helium atom 100 D.1 Perturbative treatment for Helium atom . 100 D.2 Variational treatment for Helium atom . 101 D.3 Beyond-HF treatment for Helium atom . 102 D.4 Laboratory . 104 E More about pseudopotentials 105 E.1 An early idea . 105 E.2 A modern view . 106 iii Introduction The aim of these lecture notes is to provide an introduction to methods and techniques used in the numerical solution of simple (non-relativistic) quantum- mechanical problems, with special emphasis on atomic and condensed-matter physics. The practical sessions are meant to be a sort of \computational lab- oratory", introducing the basic ingredients used in the calculation of materials properties at a much larger scale. The latter is a very important field of today's computational physics, due to its technological interest and potential applica- tions. The codes provided during the course are little more than templates. Stu- dents are expected to analyze them, to run them under various conditions, to examine their behavior as a function of input data, and most important, to interpret their output from a physical point of view. The students will be asked to extend or modify those codes, by adding or modifying some functionalities. For further insight on the theory of Quantum Mechanics, many excellent textbooks are available (e.g. Griffiths, Schiff, or the ever-green Dirac and Lan- dau). For further insight on the properly computational aspects of this course, we refer to the specialized texts quoted in the Bibliography section, and in particular to the book of Thijssen. 0.1 About Software This course assumes some basic knowledge of how to write and execute simple programs, and how to plot their results. All that is needed is a Fortran or C compiler and some visualization software. The target machine is a PC running Linux, but you can also use a Macintosh or a Windows PC, as long as the mentioned software is installed and working, and if you know how to use it in practice. For Windows 10, the path of least resistance is to enable the Windows Subsystem for Linux (WSL-2) and to install a Linux distribution and an X window client. This gives access to a very functional Linux shell. 0.1.1 Compilers In order to run a code written in any programming language, we must first translate it into machine language, i.e. a language that the computer can understand. The translation is done by an interpreter or by a compiler: the former translates and immediately executes each instruction, the latter takes the file, produces the so-called object code that together with other object codes 1 and with libraries is finally assembled into an executable file. Python, Java, or at an higher level, Matlab, Mathematica, are examples of \interpreted" language. Fortran, C, C++ are \compiled" languages. Our codes are written in Fortran 90 (or 95, or later). This is a sophisti- cated and complex language offering dynamical memory management, arrays operations (e.g. matrix-vector products), modular and object-based structure. Fortran 90 maintains a wide compatibility with existing Fortran 77 codes, while remaining as efficient as Fortran 77 was. It is worth mentioning that the first applications of computers to physics date back to well before the birth of mod- ern computer languages like C++, python, or even C: there is a large number of codes and libraries written in Fortran 77 (or even Fortran 66!) and still widely used in physics. Even among physicists, however, Fortran is no longer as common and widespread as it used to be, but online resources are still easy to find1. The codes themselves are very simple and make little usage of advanced language features. In any case, there is no problem if a student prefers to use a more widespread language like C/C++. A version of all codes in C is also available, with no warranty about the quality of the C code in terms of elegance and good coding practice. In all cases, you need a C or Fortran compiler. The C compiler gcc is free and can be installed on all operating systems (in Linux PCs it is always present). Less-then-archaic versions of gcc include a Fortran compiler, called gfortran. 0.1.2 Visualization Tools Visualization of data produced by the codes (wave functions, charge densities, various other quantities) has a central role in the analysis and understanding of the results. Code gnuplot can be used to make two-dimensional or three- dimensional plots of data or of analytical expressions. gnuplot is open-source software, available for all operating systems and often found pre-installed on Linux PCs. An introduction to gnuplot, with many links to more resources, can be found in http://www.gnuplot.info/help.html. Another software that can be used is Grace2, formerly known as xmgr. This is also open-source and highly portable, has a graphical user interface and thus it is easier to use than gnuplot, whose syntax is not always easy to remember. 0.1.3 Mathematical Libraries The usage of efficient mathematical libraries is crucial in \serious" calculations. Some of the codes use routines from the BLAS3 (Basic Linear Algebra Subpro- grams) library and from LAPACK4 (Linear Algebra PACKage).
Load more

Recommended publications
  • CUDA 6 and Beyond
    MUMPS USERS DAYS JUNE 1ST / 2ND 2017 Programming heterogeneous architectures with libraries: A survey of NVIDIA linear algebra libraries François Courteille |Principal Solutions Architect, NVIDIA |[email protected] ACKNOWLEDGEMENTS Joe Eaton , NVIDIA Ty McKercher , NVIDIA Lung Sheng Chien , NVIDIA Nikolay Markovskiy , NVIDIA Stan Posey , NVIDIA Steve Rennich , NVIDIA Dave Miles , NVIDIA Peng Wang, NVIDIA Questions: [email protected] 2 AGENDA Prolegomena NVIDIA Solutions for Accelerated Linear Algebra Libraries performance on Pascal Rapid software development for heterogeneous architecture 3 PROLEGOMENA 124 NVIDIA Gaming VR AI & HPC Self-Driving Cars GPU Computing 5 ONE ARCHITECTURE BUILT FOR BOTH DATA SCIENCE & COMPUTATIONAL SCIENCE AlexNet Training Performance 70x Pascal [CELLR ANGE] 60x 16nm FinFET 50x 40x CoWoS HBM2 30x 20x [CELLR ANGE] NVLink 10x [CELLR [CELLR ANGE] ANGE] 0x 2013 2014 2015 2016 cuDNN Pascal & Volta NVIDIA DGX-1 NVIDIA DGX SATURNV 65x in 3 Years 6 7 8 8 9 NVLINK TO CPU IBM Power Systems Server S822LC (codename “Minsky”) DDR4 DDR4 2x IBM Power8+ CPUs and 4x P100 GPUs 115GB/s 80 GB/s per GPU bidirectional for peer traffic IB P8+ CPU P8+ CPU IB 80 GB/s per GPU bidirectional to CPU P100 P100 P100 P100 115 GB/s CPU Memory Bandwidth Direct Load/store access to CPU Memory High Speed Copy Engines for bulk data movement 1615 UNIFIED MEMORY ON PASCAL Large datasets, Simple programming, High performance CUDA 8 Enable Large Oversubscribe GPU memory Pascal Data Models Allocate up to system memory size CPU GPU Higher Demand
    [Show full text]
  • Lesson 1: Solutions to Polynomial Equations
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 1: Solutions to Polynomial Equations Classwork Opening Exercise How many solutions are there to the equation 푥2 = 1? Explain how you know. Example 1: Prove that a Quadratic Equation Has Only Two Solutions over the Set of Complex Numbers Prove that 1 and −1 are the only solutions to the equation 푥2 = 1. Let 푥 = 푎 + 푏푖 be a complex number so that 푥2 = 1. a. Substitute 푎 + 푏푖 for 푥 in the equation 푥2 = 1. b. Rewrite both sides in standard form for a complex number. c. Equate the real parts on each side of the equation, and equate the imaginary parts on each side of the equation. d. Solve for 푎 and 푏, and find the solutions for 푥 = 푎 + 푏푖. Lesson 1: Solutions to Polynomial Equations S.1 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from PreCal-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 1 M3 PRECALCULUS AND ADVANCED TOPICS Exercises Find the product. 1. (푧 − 2)(푧 + 2) 2. (푧 + 3푖)(푧 − 3푖) Write each of the following quadratic expressions as the product of two linear factors. 3. 푧2 − 4 4. 푧2 + 4 5. 푧2 − 4푖 6. 푧2 + 4푖 Lesson 1: Solutions to Polynomial Equations S.2 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds.
    [Show full text]
  • Use of Iteration Technique to Find Zeros of Functions and Powers of Numbers
    USE OF ITERATION TECHNIQUE TO FIND ZEROS OF FUNCTIONS AND POWERS OF NUMBERS Many functions y=f(x) have multiple zeros at discrete points along the x axis. The location of most of these zeros can be estimated approximately by looking at a graph of the function. Simple roots are the ones where f(x) passes through zero value with finite value for its slope while double roots occur for functions where its derivative also vanish simultaneously. The standard way to find these zeros of f(x) is to use the Newton-Raphson technique or a variation thereof. The procedure is to expand f(x) in a Taylor series about a starting point starting with the original estimate x=x0. This produces the iteration- 2 df (xn ) 1 d f (xn ) 2 0 = f (xn ) + (xn+1 − xn ) + (xn+1 − xn ) + ... dx 2! dx2 for which the function vanishes. For a function with a simple zero near x=x0, one has the iteration- f (xn ) xn+1 = xn − with x0 given f ′(xn ) For a double root one needs to keep three terms in the Taylor series to yield the iteration- − f ′(x ) + f ′(x)2 − 2 f (x ) f ′′(x ) n n n ∆xn+1 = xn+1 − xn = f ′′(xn ) To demonstrate how this procedure works, take the simple polynomial function- 2 4 f (x) =1+ 2x − 6x + x Its graph and the derivative look like this- There appear to be four real roots located near x=-2.6, -0.3, +0.7, and +2.2. None of these four real roots equal to an integer, however, they correspond to simple zeros near the values indicated.
    [Show full text]
  • 5.2 Polynomial Functions 5.2 Polynomialfunctions
    5.2 Polynomial Functions 5.2 PolynomialFunctions This is part of 5.1 in the current text In this section, you will learn about the properties, characteristics, and applications of polynomial functions. Upon completion you will be able to: • Recognize the degree, leading coefficient, and end behavior of a given polynomial function. • Memorize the graphs of parent polynomial functions (linear, quadratic, and cubic). • State the domain of a polynomial function, using interval notation. • Define what it means to be a root/zero of a function. • Identify the coefficients of a given quadratic function and he direction the corresponding graph opens. • Determine the verte$ and properties of the graph of a given quadratic function (domain, range, and minimum/maximum value). • Compute the roots/zeroes of a quadratic function by applying the quadratic formula and/or by factoring. • !&etch the graph of a given quadratic function using its properties. • Use properties of quadratic functions to solve business and social science problems. DescribingPolynomialFunctions ' polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is the product of a number, called the coefficient of the term, and a variable raised to a non-negative integer (a natural number) power. Definition ' polynomial function is a function of the form n n ) * f(x =a n x +a n ) x − +...+a * x +a ) x+a +, − wherea ,a ,...,a are real numbers andn ) is a natural number. � + ) n ≥ In a previous chapter we discussed linear functions, f(x =mx=b , and the equation of a horizontal line, y=b .
    [Show full text]
  • Factors, Zeros, and Solutions
    Mathematics Instructional Plan – Algebra II Factors, Zeros, and Solutions Strand: Functions Topic: Exploring relationships among factors, zeros, and solutions Primary SOL: AII.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Related SOL: AII.1d, AII.4b, AII.7b Materials Zeros and Factors activity sheet (attached) Matching Cards activity sheet (attached) Sorting Challenge activity sheet (attached) Graphing utility Colored paper Graph paper Vocabulary factor, fundamental theorem of algebra, imaginary numbers, multiplicity, polynomial, rate of change, real numbers, root, slope, solution, x-intercept, y-intercept, zeros of a function Student/Teacher Actions: What should students be doing? What should teachers be doing? Time: 90 minutes 1. Have students solve the equation 3x + 5 = 11. Then, have students set the left side of the equation equal to zero. Tell students to replace zero with y and consider the line y = 3x − 6. Instruct them to graph the line on graph paper. Review the terms slope, x- intercept, and y-intercept. Then, have them perform the same procedure (i.e., solve, set left side equal to zero, replace zero with y, and graph) for the equation −2x + 6 = 4. They should notice that in both cases, the x-intercept is the same as the solution to the equation. 2. Students explored factoring of zeros in Algebra I. Use the following practice problem to review: y = x2 + 4x + 3. Facilitate a discussion with students using the following questions: o “What is the first step in graphing the given line?” o “How do you find the factors of the line?” o “What role does zero play in graphing the line?” o “In how many ‘places’ did this line cross (intersect) the x-axis?” o “When the function crosses the x-axis, what is the value of y?” o “What is the significance of the x-intercepts of the graphs?” 3.
    [Show full text]
  • Elementary Functions, Student's Text, Unit 21
    DOCUMENT RESUME BD 135 629 SE 021 999 AUTHOR Allen, Frank B.; And Others TITLE Elementary Functions, Student's Text, Unit 21. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE 61 NOTE 398p.; For related documents, see SE 021 987-022 002 and ED 130 870-877; Contains occasional light type EDRS PRICE MF-$0.83 HC-$20.75 Plus Postage. DESCRIPTO2S *Curriculum; Elementary Secondary Education; Instruction; *Instructional Materials; Mathematics Education; *Secondary School Mathematics; *Textbooks IDENTIFIERS *Functions (Mathematics); *School Mathematics Study Group ABSTRACT Unit 21 in the SMSG secondary school mathematics series is a student text covering the following topics in elementary functions: functions, polynomial functions, tangents to graphs of polynomial functions, exponential and logarithmic functions, and circular functions. Appendices discuss set notation, mathematical induction, significance of polynomials, area under a polynomial graph, slopes of area functions, the law of growth, approximation and computation of e raised to the x power, an approximation for ln x, measurement of triangles, trigonometric identities and equations, and calculation of sim x and cos x. (DT) *********************************************************************** Documents acquired by ERIC include many informal unpublished * materials not available from other sources. ERIC makes every effort * * to obtain the best copy available. Nevertheless, items of marginal * * reproducibility are often encountered and this affects the quality * * of the microfiche and hardcopy reproductions ERIC makes available * * via the ERIC Document Reproduction Service (EDRS). EDRS is not * responsible for the quality of the original document. Reproductions * * supplied by EDRS are the best that can be made from the original.
    [Show full text]
  • LARGE-SCALE COMPUTATION of PSEUDOSPECTRA USING ARPACK and EIGS∗ 1. Introduction. the Matrices in Many Eigenvalue Problems
    SIAM J. SCI. COMPUT. c 2001 Society for Industrial and Applied Mathematics Vol. 23, No. 2, pp. 591–605 LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA USING ARPACK AND EIGS∗ THOMAS G. WRIGHT† AND LLOYD N. TREFETHEN† Abstract. ARPACK and its Matlab counterpart, eigs, are software packages that calculate some eigenvalues of a large nonsymmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large- scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000. Key words. Arnoldi, ARPACK, eigenvalues, implicit restarting, pseudospectra AMS subject classifications. 65F15, 65F30, 65F50 PII. S106482750037322X 1. Introduction. The matrices in many eigenvalue problems are too large to allow direct computation of their full spectra, and two of the iterative tools available for computing a part of the spectrum are ARPACK [10, 11]and its Matlab counter- part, eigs.1 For nonsymmetric matrices, the mathematical basis of these packages is the Arnoldi iteration with implicit restarting [11, 23], which works by compressing the matrix to an “interesting” Hessenberg matrix, one which contains information about the eigenvalues and eigenvectors of interest. For general information on large-scale nonsymmetric matrix eigenvalue iterations, see [2, 21, 29, 31]. For some matrices, nonnormality (nonorthogonality of the eigenvectors) may be physically important [30].
    [Show full text]
  • Applications of Derivatives
    5128_Ch04_pp186-260.qxd 1/13/06 12:35 PM Page 186 Chapter 4 Applications of Derivatives n automobile’s gas mileage is a function of many variables, including road surface, tire Atype, velocity, fuel octane rating, road angle, and the speed and direction of the wind. If we look only at velocity’s effect on gas mileage, the mileage of a certain car can be approximated by: m(v) ϭ 0.00015v 3 Ϫ 0.032v 2 ϩ 1.8v ϩ 1.7 (where v is velocity) At what speed should you drive this car to ob- tain the best gas mileage? The ideas in Section 4.1 will help you find the answer. 186 5128_Ch04_pp186-260.qxd 1/13/06 12:36 PM Page 187 Section 4.1 Extreme Values of Functions 187 Chapter 4 Overview In the past, when virtually all graphing was done by hand—often laboriously—derivatives were the key tool used to sketch the graph of a function. Now we can graph a function quickly, and usually correctly, using a grapher. However, confirmation of much of what we see and conclude true from a grapher view must still come from calculus. This chapter shows how to draw conclusions from derivatives about the extreme val- ues of a function and about the general shape of a function’s graph. We will also see how a tangent line captures the shape of a curve near the point of tangency, how to de- duce rates of change we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point.
    [Show full text]
  • Optimization Algorithms on Matrix Manifolds
    00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Chapter Six Newton’s Method This chapter provides a detailed development of the archetypal second-order optimization method, Newton’s method, as an iteration on manifolds. We propose a formulation of Newton’s method for computing the zeros of a vector field on a manifold equipped with an affine connection and a retrac­ tion. In particular, when the manifold is Riemannian, this geometric Newton method can be used to compute critical points of a cost function by seeking the zeros of its gradient vector field. In the case where the underlying space is Euclidean, the proposed algorithm reduces to the classical Newton method. Although the algorithm formulation is provided in a general framework, the applications of interest in this book are those that have a matrix manifold structure (see Chapter 3). We provide several example applications of the geometric Newton method for principal subspace problems. 6.1 NEWTON’S METHOD ON MANIFOLDS In Chapter 5 we began a discussion of the Newton method and the issues involved in generalizing such an algorithm on an arbitrary manifold. Sec­ tion 5.1 identified the task as computing a zero of a vector field ξ on a Riemannian manifold equipped with a retraction R. The strategy pro­ M posed was to obtain a new iterate xk+1 from a current iterate xk by the following process. 1. Find a tangent vector ηk Txk such that the “directional derivative” of ξ along η is equal to ∈ ξ.
    [Show full text]
  • Lesson 11 M1 ALGEBRA II
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M1 ALGEBRA II Lesson 11: The Special Role of Zero in Factoring Student Outcomes . Students find solutions to polynomial equations where the polynomial expression is not factored into linear factors. Students construct a polynomial function that has a specified set of zeros with stated multiplicity. Lesson Notes This lesson focuses on the first part of standard A-APR.B.3, identifying zeros of polynomials presented in factored form. Although the terms root and zero are interchangeable, for consistency only the term zero is used throughout this lesson and in later lessons. The second part of the standard, using the zeros to construct a rough graph of a polynomial function, is delayed until Lesson 14. The ideas that begin in this lesson continue in Lesson 19, in which students will be able to associate a zero of a polynomial function to a factor in the factored form of the associated polynomial as a consequence of the remainder theorem, and culminate in Lesson 39, in which students apply the fundamental theorem of algebra to factor polynomial expressions completely over the complex numbers. Classwork Scaffolding: Here is an alternative opening Opening Exercise (12 minutes) activity that may better illuminate the special role of Opening Exercise zero. ퟐ ퟐ Find all solutions to the equation (풙 + ퟓ풙 + ퟔ)(풙 − ퟑ풙 − ퟒ) = ퟎ. For each equation, list some possible values for 푥 and 푦. The main point of this opening exercise is for students to recognize and then formalize that the statement “If 푎푏 = 0, then 푎 = 0 or 푏 = 0” applies not only when 푎 and 푏 are 푥푦 = 10, 푥푦 = 1, numbers or linear functions (which we used when solving a quadratic equation), but also 푥푦 = −1, 푥푦 = 0 applies to cases where 푎 and 푏 are polynomial functions of any degree.
    [Show full text]
  • Real Zeros of Polynomial Functions
    333353_0200.qxp 1/11/07 2:00 PM Page 91 Polynomial and Chapter 2 Rational Functions y y y 2.1 Quadratic Functions 2 2 2 2.2 Polynomial Functions of Higher Degree x x x −44−2 −44−2 −44−2 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra Polynomial and rational functions are two of the most common types of functions 2.6 Rational Functions and used in algebra and calculus. In Chapter 2, you will learn how to graph these types Asymptotes of functions and how to find the zeros of these functions. 2.7 Graphs of Rational Functions 2.8 Quadratic Models David Madison/Getty Images Selected Applications Polynomial and rational functions have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. ■ Automobile Aerodynamics, Exercise 58, page 101 ■ Revenue, Exercise 93, page 114 ■ U.S. Population, Exercise 91, page 129 ■ Impedance, Exercises 79 and 80, page 138 ■ Profit, Exercise 64, page 145 ■ Data Analysis, Exercises 41 and 42, page 154 ■ Wildlife, Exercise 43, page 155 ■ Comparing Models, Exercise 85, page 164 ■ Media, Aerodynamics is crucial in creating racecars.Two types of racecars designed and built Exercise 18, page 170 by NASCAR teams are short track cars, as shown in the photo, and super-speedway (long track) cars. Both types of racecars are designed either to allow for as much downforce as possible or to reduce the amount of drag on the racecar. 91 333353_0201.qxp 1/11/07 2:02 PM Page 92 92 Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions The Graph of a Quadratic Function What you should learn ᭿ Analyze graphs of quadratic functions.
    [Show full text]
  • Root-Finding 3.1 Physical Problems
    Physics 200 Lecture 3 Root-Finding Lecture 3 Physics 200 Laboratory Monday, February 14th, 2011 The fundamental question answered by this week's lab work will be: Given a function F (x), find some/all of the values xi for which F (xi) = 0. It's a f g modest goal, and we will use a simple method to solve the problem. But, as we shall see, there are a wide range of physical problems that have, at their heart, just such a question. We'll start in the simplest, polynomial setting, and work our way up to the \shooting" method. 3.1 Physical Problems We'll set up some direct applications of root-finding with familiar physical examples, and then shift gears and define a numerical root-finding routine that can be used to solve a very different set of problems. 3.1.1 Orbital Motion In two-dimensions, with a spherically symmetric potential (meaning here that V (x; y; z) = V (r), a function of a single variable, r x2 + y2 + z2, ≡ the distance to the origin) we can use circular coordinates to write the total p energy of a test particle moving under the influence of this potential as 1 E = m r_2 + r2 φ_2 + V (r): (3.1) 2 Conservation of momentum tells us that the z-component of angular mo- 2 mentum is conserved, with Lz = (r p) = m r φ_, so we can rewrite the × z 1 of 13 3.1. PHYSICAL PROBLEMS Lecture 3 energy as: 1 1 L2 E = m r_2 + z + V (r) (3.2) 2 2 m r2 ≡U(r) where U(r) defines an “effective potential"| { we{z have} turned a two-dimensional problem into a one-dimensional problem for the coordinate r, and an effec- tive potential that governs the motion in this setting.
    [Show full text]