found between x and x + dx is then equal to j^(x)[ dx, provided that tjkx) is normalized in the usual way: X - I (6.1)
|
|
- Johnathan Dawson
- 7 years ago
- Views:
Transcription
1 VI. OPERATORS & EXPECTATION VALUES 1. The Calculation of Average Values The description of quantum states in terms of probability amplitudes at once implies the possibility of using these probabi lity amplitudes to calculate the average values of various quanti ties for systems that are in a a specified state. Consider, for example, a bound state of a particle in a one-dimensional poten tial. Such a state has a unique value of the energy,'but there is a continuous distribution of the possible values of the position, x. The probability amplitude associated with a particular value of x is given by 4>(x). The probability that the particle will be! I 2 found between x and x + dx is then equal to j^(x)[ dx, provided that tjkx) is normalized in the usual way: X - I (6.1) If we ask "What is the average position of a particle described by ikx)?"- w^ can supply the answer by evaluating a weighted mean value, just as in classical probability calculations. We multiply each particular value of x by the probability of having that value, and sum (integrate) over all possibilities: *, Such averages are called expectation values in quantum mechanics. This name vividly expresses the essentially statistical nature o:5 ; -1-
2 -2- the quantum-mechanical description. One can imagine a huge number of identical systems, each consisting of a particle in the state described by ip(x). If one carried out a set of experiments to determine the location of each of these particles, the results would be distributed in accordance with the relative 2 The result of an individual observation is not predictable, but the mean of all the measurements is well defined. It is customary to denote the expectation value of any quantity Q by enclosing it within angular brackets ; thus we put Expectation value of x (6.2) 1 To take a specific example, consider any one bound state of a particle in a "violin-string" state: VI The expectation value of x for this state is then given by L? C ' 2- t J X Sm O It is easy to verify that the result of this calculation is,that (x^ is equal to L/2 for any n. Notice that the expectation value of a quantity is in principle not at all the same thing as the most probable value. For example, if we are concerned with measurements of position, the most probable value of x is the value / Iffi
3 -3- at which J«y(x) ( is greatesti If we consider the violin-string states u/(x), then for n = 1 the expectation value and the most probable value do coincide, but for n = 2 the value of is zero; thus a particle in this state would never be found at a value of x precisely equal to C^. *N Given that I MX) j dx is the probability associated with a particular value of x, we can use it to calculate the expecta tion value of any function of x.. We simply have <fm> - Thus we can, for example, find the mean squared value of x 2 2. Expectation Values involving Operators Suppose that we again consider any individual energy-state of a particle in a one-dimensional box, and this time ask what is the expectation value of the linear momentum p. A consideration j^ of the symmetry of the situation suggests, without the need for any calculation, that the answer is zero; the particle is equally likely to be found moving to right or left. We can spell this out more formally by noting that the wave function can be written as a superposition, with equal weights, of plane-wave amplitudes belonging to equal and opposite values of the wave-number k : m km x - fit - ne I yv rt /\ Thus k = + k, and the momentum p (=ftk) is equal to + ^k n *x ^ n This makes it clear that the probability associated with each
4 4 value of the two values of p is 1/2, and so we have <px> = [Notice, again, the distinction between expectation value and most probable value. Individual measurements on this state would never yield a value of k equal to zero, but only + k.] The above calculation is very simple and straightforward, but this is mainly because the situation itself is very simple. What we shall now do is to introduce a method of using the Schrfidinger amplitude itself to calculate ^p > and related quantities. With its help we can then handle problems that are far less obvious than the one we are now using as an example. The starting point is the following very important concept: In quantum mechanics we can associate various dynamical quantities with particular mathematical operators. This is a rather general statement, but we shall at once make it explicit in connection with linear momentum. We have seen that the probability amplitude for a free particle of given momentum v is of the form /1 t(k*~ - ft e where k = p /fi. Thus we can put or We then propose that, in general, the operator *~t*\c0fbty r operating
5 -5- on a Schrodinger amplitude ip, is equivalent to multiplying by p. Thus we put fi (6-3) We then proceed to evaluate the expectation value of p by means of the following equation: < (6.4) If, as in any one of the bound states of a particle in a box, the SchrOdinger amplitude $ embodies contributions from more than one value of ' p, the mathematical operator corresponding to p, *k «V applied to J, yields all these component momenta and the integral in eq. (6.4) is an automatic prescription for forming the appropriate weighted average. It is important to observe the order of the various factors in this integral. Now that p is represented j^> by a differential operator/ applying to anything that follows it, we must place the factor $* to its left so as to make clear that it is not subject to this operation. Suppose once again that ip represents a particular energy- state of a particle in a one-dimensional box: Then 2. L o
6 -6- But the functions sin k x and cos k x are orthogonal (i.e., j sin k x cos k x dx = 0), so this calculation gives vp / = 0 0 J n n ^ N ^x as we know it must. However, a calculation of p by the same technique gives a non-zero result. The operator correspending to p jf± is -# ( d /^x ), and so for this same state we have ^ Nr 2 w 2 ' f (sin ' knx)[^/sin fv\ krx) dx / \. ty, 2 i.e., <px > = V kn 2 f L 2 L T n J f sin k n x dx 0 This is just what we expect, since the (kinetic) energy E of the state will be given by <P 2 > F n n 2m 2m With k = ntt/l, this gives us the familiar result 2 2 n h n 8mL 2 We could, of course, have obtained this expectation value of p wrv much more readily by going back to the description of the state as a. 50:50 mixture of the two momentum components Ik. On this basis we have simply <PX2 > = K 2 <k2 > = # 2 [ i(kn ) 2 + i(-kn ) 2] = Thus, up to this point, our use of the momentum operator technique has been like the proverbial sledge-hcmmer to kill a fly. But let us now take an example where it is not. so trivial.
7 -7- Consider a particle in one of the characteristic energy states in a harmonicoscillator potential. Now the value of p, as defined x-0 by + /2m[E - V(x)J, takes on an infinite number of different values (including imaginary ones). Given a knowledge of 1 /, we can however evaluate with the help of eq. (6.4). Suppose that the state in question is the lowest state of the oscillator The spatial factor in its wave function is of the form = A e 2/0 2 -x /2a (6.5a) where 2 a =s mc [cf. Notes, Ch. Ill, pp. 19 ff.] The normalization of \ / leads to the condition The value- of A2 = a ATT ) is given by (6.5b) <PX> - -in K A *2 J C e -* / /2a = %- j» "CO 2~, 2 - x /a dx d_ dx Since the integrand is the product of an odd function (x itself) with an even function, the integral between the limits + oo i s zero. This is another result that could have been expected.
8 -8- y 2 v More interesting, perhaps, is the calculation of \p >. For * this we have <px2 > = -*2A2 f v /Oa inl v /Oa X//:d U,^A/Zcl x ^^ dx ^2A2 f -x 2/a e ' dx - A' 4 (30 o 2 y 2 2 -x /a, x e 7 dx 2a 2 Substituting the value of A from eq.(6.5b), this gives us <PX 2 > = - (6.6) 2a The calculation of expectation values of the energy E can likewise be based on the use of an equivalent operator. Referring again in the first instance to a free-particle wave function, we have and hence TV i (kx-et/fi) = Ae N ' or E <^-> ih ^ (6.7) ^t If, now, we take a wave-function that represents a superposition of different energy-states, we have e-^n*/* ifl Then E = C f /f \/ _ 7 ' ' E
9 -9- If the different ( f s are orthogonal and normalized, this reduces to the equation < E >= ra n Thus/ again as one might expect/ the value of ^E^ is just a weighted average of the energies of the component states, with weighting factors equal to the integrated probabilities associated with the individual components. In some circumstances it may be of interest to calculate the mean values of the kinetic energy and the potential energy separately for a particle in a given state. The ways of evaluat ing these quantities are essentially contained in what we have already done. The operator for the kinetic energy K is the 2 operator corresponding to p /2m ; thus for one-dimensional problems we put K ~~ (6-8a) and for three dimensions we have In a one-dimensional system we therefore have K <r~> -_ (6.8b) For the mean value of the potential energy/ we simply use the procedure that applies to an arbitrary function of position, so that V(x) I//" <^x (6.10)
10 -10 If we consider the (one-dimensional) Schrodinger equation as it applies to an individual energy-state, we have dx -* Multiplying by \^ and integrating over all x then leads, with the help of eqs. (6.9) and (6.10), to the reasonable result <K> + <V> = E (6.11) 3. Variances and Uncertainty Relations In statistical analysis, when one is dealing with a quantity that has a certain distribution of values, an automatic measure of the spread of the distribution around its mean value is provided by the standard deviation, CT. The square of CT, known as the variance, is the mean squared deviation of the individual values from the mean. Thus for some arbitrary quantity Q, we have Variance = -CT 2 = (Q - <Q> ) 2 >C ClV 4 But (Q - <Q> ) 2 = Q 2-2Q <Q> + <Q> 2, and the variance is the expectation value of this expression. This gives <rq2 = <Q2 > - 2 <Q><Q> 4- <Q> 2 * i.e., <TQ2 = <Q2 > - <Q> 2 (6.12) tf Only if Q is limited to a single sharp value does the variance 2 drop to zero; in all other cases the difference between t and Q is positive.
11 -11- We can use this measure of the width of a probability distribution to examine the uncertainty relations in more specific terms. Consider, for example, the lowest state of the harmonic oscillator, as described by eqs.(6.5). For this we have = o fir a 3 ^ 2, Thus CT 2 Now in Section 2 we calculated the values of p > and J\. the results were <PX> =o, 2, -h 2 <PV > = 2 a' 2 Hence h 2 2 a 2 It follows, then, that the following relation holds: x Px = ^ (6.13) This uncertainty product is thus independent of the parameter a that characteri2es the width of the position probability distribu tion. It happers that the value fc/2, characteristic of these
12 -12- Gaussian error-function probability distributions, represents the smallest achievable value of Ap «/\ x, where A p and Ax are defined as the standard deviations of the probability distributions in question. [4. The Quantum-Mechanical Equivalent of Newton's Law An interesting application of the calculation of expecta tion values is the formulation of an equation that parallels the basic law of motion in classical dynamics. We begin with the expectation value of p as defined in eq. (6.4), and then con- «sider its time derivative: Now with the help of the time-dependent Schrfldinger equation we can convert the integrands on the right into forms that do not explicitly involve t:. w it *
13 -13- Hence at -JL 2.w fi J ^ For any reasonable wave- function, the values of \ and its derivatives vanish at x = + cx> Thus we are left with the result Since the negative gradient of the potential energy is equal to the force derived from that potential, eg. (6.14) corresponds to a statement of F = dp/dt in terms of the expectation values of these quantities. If one wants to carry the analysis a stage further back, one can show by similar methods that the following result also holds: m ft < x> = <' px>
14 Expectation Values in Superpositions of States If we have a superposition of two different energy-states of a quantum-mechanical system, there is a harmonic time-dependence of the expectation values of position and related quantities. Suppose that the state in question is given by. Then + f 2-f./il/,^x) VMd*} cos cot where ^0 = (E 2 - Ej)/K Each of the three integrals in the above expression is a definite integral representing a quantity of the dimension of length. Thus the equation for ^x> can be written in the simplified form <x> = A + B cos 60 1 (6.16) If the wave-function in question describes an electrically charged particle, one can see here the basis of a picture of an oscillating electric dipole, with the implied possibility of associated v radiation characterized by the frequency 0).
Time dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationQuantum Mechanics: Postulates
Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
More information7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.
7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated
More informationTheory of electrons and positrons
P AUL A. M. DIRAC Theory of electrons and positrons Nobel Lecture, December 12, 1933 Matter has been found by experimental physicists to be made up of small particles of various kinds, the particles of
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationReview D: Potential Energy and the Conservation of Mechanical Energy
MSSCHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Fall 2005 Review D: Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Non-conservative Force... 2 D.1.1 Introduction...
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationChapter 15 Collision Theory
Chapter 15 Collision Theory 151 Introduction 1 15 Reference Frames Relative and Velocities 1 151 Center of Mass Reference Frame 15 Relative Velocities 3 153 Characterizing Collisions 5 154 One-Dimensional
More informationTopic 3b: Kinetic Theory
Topic 3b: Kinetic Theory What is temperature? We have developed some statistical language to simplify describing measurements on physical systems. When we measure the temperature of a system, what underlying
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationLecture L22-2D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More informationarxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationApplications of Second-Order Differential Equations
Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationarxiv:physics/0004029v1 [physics.ed-ph] 14 Apr 2000
arxiv:physics/0004029v1 [physics.ed-ph] 14 Apr 2000 Lagrangians and Hamiltonians for High School Students John W. Norbury Physics Department and Center for Science Education, University of Wisconsin-Milwaukee,
More informationHow do we obtain the solution, if we are given F (t)? First we note that suppose someone did give us one solution of this equation
1 Green s functions The harmonic oscillator equation is This has the solution mẍ + kx = 0 (1) x = A sin(ωt) + B cos(ωt), ω = k m where A, B are arbitrary constants reflecting the fact that we have two
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationAim : To study how the time period of a simple pendulum changes when its amplitude is changed.
Aim : To study how the time period of a simple pendulum changes when its amplitude is changed. Teacher s Signature Name: Suvrat Raju Class: XIID Board Roll No.: Table of Contents Aim..................................................1
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationLinear Equations and Inequalities
Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................
More information5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM
5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationLet s first see how precession works in quantitative detail. The system is illustrated below: ...
lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationAssessment Plan for Learning Outcomes for BA/BS in Physics
Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationDIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents
DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition
More informationARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationIntroduction to Schrödinger Equation: Harmonic Potential
Introduction to Schrödinger Equation: Harmonic Potential Chia-Chun Chou May 2, 2006 Introduction to Schrödinger Equation: Harmonic Potential Time-Dependent Schrödinger Equation For a nonrelativistic particle
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationSettling a Question about Pythagorean Triples
Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:
More information1. Degenerate Pressure
. Degenerate Pressure We next consider a Fermion gas in quite a different context: the interior of a white dwarf star. Like other stars, white dwarfs have fully ionized plasma interiors. The positively
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More informationCHAPTER 2. Eigenvalue Problems (EVP s) for ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationarxiv:1603.01211v1 [quant-ph] 3 Mar 2016
Classical and Quantum Mechanical Motion in Magnetic Fields J. Franklin and K. Cole Newton Department of Physics, Reed College, Portland, Oregon 970, USA Abstract We study the motion of a particle in a
More informationLeast-Squares Intersection of Lines
Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationThe Point-Slope Form
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationPLANNING PROBLEMS OF A GAMBLING-HOUSE WITH APPLICATION TO INSURANCE BUSINESS. Stockholm
PLANNING PROBLEMS OF A GAMBLING-HOUSE WITH APPLICATION TO INSURANCE BUSINESS HARALD BOHMAN Stockholm In the classical risk theory the interdependence between the security loading and the initial risk reserve
More informationLecture 8. Generating a non-uniform probability distribution
Discrete outcomes Lecture 8 Generating a non-uniform probability distribution Last week we discussed generating a non-uniform probability distribution for the case of finite discrete outcomes. An algorithm
More information19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style
Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have
More information2.6 The driven oscillator
2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More information1. First-order Ordinary Differential Equations
Advanced Engineering Mathematics 1. First-order ODEs 1 1. First-order Ordinary Differential Equations 1.1 Basic concept and ideas 1.2 Geometrical meaning of direction fields 1.3 Separable differential
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More information6 J - vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion
More informationFree Electron Fermi Gas (Kittel Ch. 6)
Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)
More informationDO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS
DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS Quantum Mechanics or wave mechanics is the best mathematical theory used today to describe and predict the behaviour of particles and waves.
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationSolutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 7
Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid April 21, 2002 Chapter 7 Problem 7.2 Obtain the Lorentz transformation in which the velocity is at an infinitesimal angle
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More information3. Electronic Spectroscopy of Molecules I - Absorption Spectroscopy
3. Electronic Spectroscopy of Molecules I - Absorption Spectroscopy 3.1. Vibrational coarse structure of electronic spectra. The Born Oppenheimer Approximation introduced in the last chapter can be extended
More information