Ph12b Solution Set 4
|
|
- Juliet Clark
- 5 years ago
- Views:
Transcription
1 Ph Solution Set 4 Feruary 5, 00 The state ψ S un E is transforme to p ψ S un E + p a 0 S γ E + S η E Since γ E an η E aren t mutually orthogonal, we can t use this equation irectly to compute the new ensity operator. Instea, we must change to an orthonormal system, as suggeste in the hint. One way to o this is to keep γ E an to take δ E as the secon asis vector, where δ E is etermine y η E ɛ γ E + ɛ ɛ δ E Sustituting into, we get that the new state is p ψ S un E + pa 0 S + ɛ S γ E + p ɛ ɛ S δ E 3 Rememer that if we have a state ψ i S f i E i with f i E normalize an mutually orthogonal, then we calculate the ensity operator y the formula ˆρ i ψ i S ψ i S Working in the asis 0 S, S of S, we have p ψ S a p a p a 0 S + ɛ S p ɛ p ɛ ɛ S p 0 ɛ ɛ
2 so from 3 we get that the new ensity operator is ˆρ p a a + p a ɛ a ɛ a pɛ a pɛa Comparing this to the original ensity operator ˆρ ψ S ψ S a a a a a lets us rea off that λ pɛ. + pɛ ɛ 0 0 a We egin y analysing the evolution of a single term of the form ψ ψ. If at time t we have ψt ψt, then at time t + t we have ψt + t ψt + t ψt iωt e e ψt ψt + iωt ψt e e ψt ψt iωt e e ψt ψt + iωt ψt ψt e e + ω t e e ψt ψt e e Notice that the right han sie is linear in ψt ψt. Since a ensity operator is a linear comination of ψ i t ψ i t terms, y summing the previous equation over such terms, for an aritrary ensity operator ˆρt we have ˆρt + t ˆρt iωt e e ˆρt + iωtˆρt e e + ω t e e ˆρt e e Now taking the limit of ˆρt+t ˆρt t as t 0 gives ˆρ iω e e ˆρ + iω ˆρ e e 4 t as require. In the asis { g, e }, we have e 0 so e e an Eq. 4 ecomes ρgg ρ iω ge t ρ eg ρ ee 0 ρ eg ρ ee 0 iωρge iωρ eg 0 + iω Componentwise, we get the 4 ifferential equations t ρ gg 0 t ρ ge iωρ ge t ρ eg iωρ eg t ρ ee 0 0 ρ eg ρ ee 0
3 The solutions of these equations, in terms of initial conitions at t 0, are ρ gg t ρ gg 0 ρ ge t e iωt ρ ge 0 ρ eg t e iωt ρ eg 0 ρ ee t ρ ee 0 so in terms of components of ˆρ0, ˆρt is given y ρgg 0 e ˆρt iωt ρ ge 0 e iωt ρ eg 0 ρ ee 0 c Using the same approach as in part a, if at time t we ha a pure state ˆρt ψt ψt, then at time t + t we have the ensity operator ˆρt + t g g ψt + Γt e e ψt ψt g g + Γt ψt e e + Γt g e ψt Γt ψt e g which can e rewritten as ˆρt + t g g ˆρt g g + Γt e e ˆρt g g + Γt g g ˆρt e e + Γt e e ˆρt e e + Γt g e ˆρt e g Moreover, since this equation is linear in ˆρt, it will still hol for an aritrary mixture of states. Since g, e form an orthonormal asis of the atom s Hilert space, we have g g + e e which we use with the previous equation to get ˆρt + t ˆρt t Γt Γt e e ˆρt g g + g g ˆρt e e t t Γ e e ˆρt e e + Γ g e ˆρt e g an taking the limit t 0, this ecomes ρ t Γ e e ˆρ g g Γ g g ˆρ e e Γ e e ˆρ e e + Γ g e ˆρ e g Again using g g + e e, this can e simplifie to ρ t Γ e e ˆρ Γ ˆρ e e + Γ g e ˆρ e g 5 proving the atom s master equation. 3
4 Since g 0 an e 0, we can rewrite Eq. 5 in the matrix form as t ρ eg ρ ee Γ Γ 0 ρ eg ρ ee 0 ρgg ρ + Γ ge ρ eg ρ ee Γρee Γ ρ ge Γ ρ eg Γρ ee 0 Componentwise, we get the 4 ifferential equations t ρ gg Γρ ee t ρ ge Γ ρ ge t ρ eg Γ ρ eg t ρ ee Γρ ee ρ eg ρ ee 0 The one for ρ gg is couple with ρ ee, so we start y solving the other three, ρ ge t e Γt/ ρ ge 0 ρ eg t e Γt/ ρ eg 0 ρ ee t e Γt ρ ee 0 Now integrating the equation for ρ gg /t from 0 to t we get ρ gg t ρ gg 0 + e Γt ρ ee 0 so in terms of components of ˆρ0, ˆρt is given y ρgg 0 + e ˆρt Γt ρ ee 0 e Γt/ ρ ge 0 e Γt/ ρ eg 0 e Γt ρ ee 0 e Comining the two ifferential equations gives a new ifferential equation, an we can comine them at the componentwise level, getting t ρ gg Γρ ee t ρ ge Γ + iω ρ ge t ρ eg Γ iω ρ eg t ρ ee Γρ ee 4
5 As in part, we start y solving the uncouple ifferential equations, getting ρ ge t e Γ +iωt ρ ge 0 ρ eg t e Γ iωt ρ eg 0 ρ ee t e Γt ρ ee 0 after which we integrate the equation for ρ gg, getting ρ gg t ρ gg 0 + e Γt ρ ee 0 To summarize, in terms of components of ˆρ0, ˆρt is given y ρ ˆρt gg 0 + e Γt ρ ee 0 e Γ +iωt ρ ge 0 e Γ iωt ρ eg 0 e Γt ρ ee 0 3 a Since we have / proaility for each of the states ψ H, ψ T, the ensity operator is ˆρ ψ H ψ H + ψ T ψ T 0 + / / / 0 / 3/4 /4 /4 /4 Eigenvalues λ of ˆρ satisfy the characteristic equation which has the solutions 3 4 λ 4 λ λ ± ± 8 6 c From the equation 3/4 λ± /4 u± 0 /4 /4 λ ± v ± we get u ± + v ± 0. If we focus on normalize eigenvectors, u ± + v±, an comining these two equations gives, after some algera, u± ± / v ± ± / 5
6 By comparing these with the values of cosπ/8, sinπ/8 which you can calculate using the half-angle formulas, you may notice that u+ cosπ/8 v + sinπ/8 u sinπ/8 v cosπ/8 Another reason to suspect that the angle π/8 woul have something to o with the solution is just y noticing that ψ H, ψ T are vectors with phases 0 an π/4, an since they contriute equally to the ensity operator, it is to e expecte that the ensity operator has some sort of symmetry aroun the axis with angle π/8. 4 a Let M e the matrix ψ a N a,. By the singular value ecomposition, there exist unitary matrices U u ai N a,i an V v j N j, such that M UDV with D ij N i,j a iagonal real N N matrix with nonnegative eigenvalues. In componentwise notation, M UDV ecomes ψ a u ai ij v j i,j which we can use to express ψ as ψ a, a,i,j, ψ a e a f u ai ij v j e a f N N ij u ai e a v j f i,j a Now we use the conition that U an V are unitary. Rememer that unitary matrices correspon to transformations from one orthonormal asis to another. Treating U as a unitary matrix acting on H A, we see that e i u ai e a a efines a new orthonormal asis of H A, an treating V as a unitary matrix acting on H B, we see that f j 6 v j f
7 efines a new orthonormal asis of H B note that here the summation is in the secon inex of v j, so this transform actually correspons to V T, ut that is irrelevant ecause the transpose of a unitary matrix is again unitary. With these new ases for H A an H B, the state ψ is represente as ψ ij e i f j i,j But rememer that D is a iagonal real matrix with nonnegative eigenvalues! Thus, ij 0 for i j. Denote also p i ii. Then the previous equation ecomes ψ pi e i f i 7 i Since we know that the state is normalize, ψ ψ, which can, y the formula given in the prolem, e expresse componentwise as p i i Thus, N i p i. From Eq. 7, since f i are orthonormal vectors, we get ˆρ A p i e i e i 8 i Since { e i } form an orthonormal asis, in that asis ˆρ a is a iagonal matrix, p p 0 ˆρ A p N Analogously, the ensity operator for system B can e written in asis { f i } as p 0 0 ˆρ B p i f i f i 0 p 0. i p N c Since from part we have representations of ˆρ A an ˆρ B as iagonal matrices, their eigenvalues are just the elements on the iagonal. Thus, reaing off from Eqs. 9 an 0 we see that ˆρ A an ˆρ B have the same eigenvalues p, p,..., p N. 7
1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationFree rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012
Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate
More informationDesigning Information Devices and Systems II Spring 2018 J. Roychowdhury and M. Maharbiz Discussion 2A
EECS 6B Designing Information Devices an Systems II Spring 208 J. Roychowhury an M. Maharbiz Discussion 2A Secon-Orer Differential Equations Secon-orer ifferential equations are ifferential equations of
More informationRepresentation theory of SU(2), density operators, purification Michael Walter, University of Amsterdam
Symmetry and Quantum Information Feruary 6, 018 Representation theory of S(), density operators, purification Lecture 7 Michael Walter, niversity of Amsterdam Last week, we learned the asic concepts of
More informationPhysics 504, Lecture 10 Feb. 24, Geometrical Fiber Optics. Last Latexed: February 21, 2011 at 15:11 1
Last Latexe: Feruary 1 11 at 15:11 1 Physics 54 Lecture 1 Fe. 4 11 1 Geometrical Fier Optics The wave guies consiere so far containe their fiels within conucting walls ut we know from stuying total internal
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationHomework 3 - Solutions
Homework 3 - Solutions The Transpose an Partial Transpose. 1 Let { 1, 2,, } be an orthonormal basis for C. The transpose map efine with respect to this basis is a superoperator Γ that acts on an operator
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationQuantum Mechanics Basic Principles
What is quantum mechanics? It is a framework for the evelopment of physical theories. It is not a complete physical theory in its own right. Quantum Mechanics Basic Principles QM slies y Michael A. Nielsen,
More informationDifferential Geometry of Surfaces
Differential Geometry of urfaces Jordan mith and Carlo équin C Diision, UC Berkeley Introduction These are notes on differential geometry of surfaces ased on reading Greiner et al. n. d.. Differential
More information1. At time t = 0, the wave function of a free particle moving in a one-dimension is given by, ψ(x,0) = N
Physics 15 Solution Set Winter 018 1. At time t = 0, the wave function of a free particle moving in a one-imension is given by, ψ(x,0) = N where N an k 0 are real positive constants. + e k /k 0 e ikx k,
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationHyperbolic Moment Equations Using Quadrature-Based Projection Methods
Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationRank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col
Review of Linear Algebra { E18 Hanout Vectors an Their Inner Proucts Let X an Y be two vectors: an Their inner prouct is ene as X =[x1; ;x n ] T Y =[y1; ;y n ] T (X; Y ) = X T Y = x k y k k=1 where T an
More informationDeterminant and Trace
Determinant an Trace Area an mappings from the plane to itself: Recall that in the last set of notes we foun a linear mapping to take the unit square S = {, y } to any parallelogram P with one corner at
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2
Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationChapter 2 Canonical Correlation Analysis
Chapter 2 Canonical Correlation Analysis Canonical correlation analysis CCA, which is a multivariate analysis method, tries to quantify the amount of linear relationships etween two sets of random variales,
More informationProblem set 2: Solutions Math 207B, Winter 2016
Problem set : Solutions Math 07B, Winter 016 1. A particle of mass m with position x(t) at time t has potential energy V ( x) an kinetic energy T = 1 m x t. The action of the particle over times t t 1
More informationExample Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
More informationLeChatelier Dynamics
LeChatelier Dynamics Robert Gilmore Physics Department, Drexel University, Philaelphia, Pennsylvania 1914, USA (Date: June 12, 28, Levine Birthay Party: To be submitte.) Dynamics of the relaxation of a
More informationQUANTUM MECHANICS I PHYS 516. Solutions to Problem Set # 5
QUANTUM MECHANICS I PHYS 56 Solutions to Problem Set # 5. Crossed E and B fields: A hydrogen atom in the N 2 level is subject to crossed electric and magnetic fields. Choose your coordinate axes to make
More informationVectors and Matrices
Chapter Vectors and Matrices. Introduction Vectors and matrices are used extensively throughout this text. Both are essential as one cannot derive and analyze laws of physics and physical measurements
More information6. Friction and viscosity in gasses
IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner
More informationarxiv: v1 [cond-mat.stat-mech] 9 Jan 2012
arxiv:1201.1836v1 [con-mat.stat-mech] 9 Jan 2012 Externally riven one-imensional Ising moel Amir Aghamohammai a 1, Cina Aghamohammai b 2, & Mohamma Khorrami a 3 a Department of Physics, Alzahra University,
More informationOrdinary Differential Equations: Homework 2
Orinary Differential Equations: Homework 2 M. Gameiro, J.-P. Lessar, J.D. Mireles James, K. Mischaikow January 30, 2017 2 0.1 Eercises Eercise 0.1.1. Let (X, ) be a metric space. function (in the metric
More informationMAT 545: Complex Geometry Fall 2008
MAT 545: Complex Geometry Fall 2008 Notes on Lefschetz Decomposition 1 Statement Let (M, J, ω) be a Kahler manifol. Since ω is a close 2-form, it inuces a well-efine homomorphism L: H k (M) H k+2 (M),
More information3. Mathematical Properties of MDOF Systems
3. Mathematical Properties of MDOF Systems 3.1 The Generalized Eigenvalue Problem Recall that the natural frequencies ω and modes a are found from [ - ω 2 M + K ] a = 0 or K a = ω 2 M a Where M and K are
More informationSOLUTIONS for Homework #3
SOLUTIONS for Hoework #3 1. In the potential of given for there is no unboun states. Boun states have positive energies E n labele by an integer n. For each energy level E, two syetrically locate classical
More informationKinematics of Rotations: A Summary
A Kinematics of Rotations: A Summary The purpose of this appenix is to outline proofs of some results in the realm of kinematics of rotations that were invoke in the preceing chapters. Further etails are
More informationComposition of Haar Paraproducts
Composition of Haar Paraproucts Brett D. Wick Georgia Institute of Technology School of Mathematics Analytic Spaces an Their Operators Memorial University of Newfounlan July 9-12, 2013 B. D. Wick (Georgia
More informationQuivers. Virginia, Lonardo, Tiago and Eloy UNICAMP. 28 July 2006
Quivers Virginia, Lonardo, Tiago and Eloy UNICAMP 28 July 2006 1 Introduction This is our project Throughout this paper, k will indicate an algeraically closed field of characteristic 0 2 Quivers 21 asic
More information10 Lorentz Group and Special Relativity
Physics 129 Lecture 16 Caltech, 02/27/18 Reference: Jones, Groups, Representations, and Physics, Chapter 10. 10 Lorentz Group and Special Relativity Special relativity says, physics laws should look the
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 3: Basic Quantum Mechanics 26th September 2016 School of Informatics, University of Edinburgh Resources 1. Quantum Computation and Quantum Information
More informationReflections and Rotations in R 3
Reflections and Rotations in R 3 P. J. Ryan May 29, 21 Rotations as Compositions of Reflections Recall that the reflection in the hyperplane H through the origin in R n is given by f(x) = x 2 ξ, x ξ (1)
More informationSimple Derivation of the Lindblad Equation
Simple Derivation of the Linbla Equation Philip Pearle - arxiv:120.2016v1 10-apr 2012 April 10,2012 Abstract The Linbla equation is an evolution equation for the ensity matrix in quantum theory. It is
More information8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current
Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the
More informationarxiv: v1 [math-ph] 10 Apr 2012
Simple Derivation of the Linbla Equation Philip Pearle 1, 1 Department of Physics, Hamilton College, Clinton, NY 1333 (Date: April 11, 01) The Linbla equation is an evolution equation for the ensity matrix
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More informationAnomalous Light Scattering by Topological PT -symmetric Particle Arrays: Supplementary Information
Anomalous Light Scattering by Topological PT -symmetric Particle Arrays: Supplementary Information C. W. Ling, Ka Hei Choi, T. C. Mok, Z. Q. Zhang, an Kin Hung Fung, Department of Applie Physics, The Hong
More informationDensity matrix equation for a bathed small system and its application to molecular magnets
City University of New York CUNY CUNY Acaemic Works Publications an Research Lehman College 013 Density matrix equation for a bathe small system an its application to molecular magnets Dmitry A. Garanin
More information1 Math 285 Homework Problem List for S2016
1 Math 85 Homework Problem List for S016 Note: solutions to Lawler Problems will appear after all of the Lecture Note Solutions. 1.1 Homework 1. Due Friay, April 8, 016 Look at from lecture note exercises:
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationComposition of Haar Paraproducts
Composition of Haar Paraproucts Brett D. Wick Georgia Institute of Technology School of Mathematics Chongqing Analysis Meeting IV Chongqing University Chongqing, China June 1, 2013 B. D. Wick (Georgia
More information1.3.3 Basis sets and Gram-Schmidt Orthogonalization
.. Basis sets and Gram-Schmidt Orthogonalization Before we address the question of existence and uniqueness, we must estalish one more tool for working with ectors asis sets. Let R, with (..-) : We can
More informationOn a limit theorem for non-stationary branching processes.
On a limit theorem for non-stationary branching processes. TETSUYA HATTORI an HIROSHI WATANABE 0. Introuction. The purpose of this paper is to give a limit theorem for a certain class of iscrete-time multi-type
More informationJointly continuous distributions and the multivariate Normal
Jointly continuous istributions an the multivariate Normal Márton alázs an álint Tóth October 3, 04 This little write-up is part of important founations of probability that were left out of the unit Probability
More informationCHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold
CHAPTER 1 : DIFFERENTIABLE MANIFOLDS 1.1 The efinition of a ifferentiable manifol Let M be a topological space. This means that we have a family Ω of open sets efine on M. These satisfy (1), M Ω (2) the
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationInvariant Extended Kalman Filter: Theory and application to a velocity-aided estimation problem
Invariant Extene Kalman Filter: Theory an application to a velocity-aie estimation problem S. Bonnabel (Mines ParisTech) Joint work with P. Martin (Mines ParisTech) E. Salaun (Georgia Institute of Technology)
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationLinear Algebra (Review) Volker Tresp 2017
Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.
More information3.6. Let s write out the sample space for this random experiment:
STAT 5 3 Let s write out the sample space for this ranom experiment: S = {(, 2), (, 3), (, 4), (, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)} This sample space assumes the orering of the balls
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationApplications of the Wronskian to ordinary linear differential equations
Physics 116C Fall 2011 Applications of the Wronskian to orinary linear ifferential equations Consier a of n continuous functions y i (x) [i = 1,2,3,...,n], each of which is ifferentiable at least n times.
More informationLogarithms. For example:
Math Review Summation Formulas Let >, let A, B, and C e constants, and let f and g e any functions. Then: f C Cf ) ) S: factor out constant ± ± g f g f ) ) )) ) S: separate summed terms C C ) 6 ) ) Computer
More informationSummation Formulas. Math Review. Let N > 0, let A, B, and C be constants, and let f and g be any functions. Then: S1: factor out constant
Computer Science Dept Va Tech August 005 005 McQuain WD Summation Formulas Let > 0, let A, B, and C e constants, and let f and g e any functions. Then: f C Cf ) ) S: factor out constant g f g f ) ) ))
More informationExpansion formula using properties of dot product (analogous to FOIL in algebra): u v 2 u v u v u u 2u v v v u 2 2u v v 2
Least squares: Mathematical theory Below we provide the "vector space" formulation, and solution, of the least squares prolem. While not strictly necessary until we ring in the machinery of matrix algera,
More informationSample Solutions from the Student Solution Manual
1 Sample Solutions from the Student Solution Manual 1213 If all the entries are, then the matrix is certainly not invertile; if you multiply the matrix y anything, you get the matrix, not the identity
More information5-4 Electrostatic Boundary Value Problems
11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions
More informationLet N > 0, let A, B, and C be constants, and let f and g be any functions. Then: S2: separate summed terms. S7: sum of k2^(k-1)
Summation Formulas Let > 0, let A, B, and C e constants, and let f and g e any functions. Then: k Cf ( k) C k S: factor out constant f ( k) k ( f ( k) ± g( k)) k S: separate summed terms f ( k) ± k g(
More informationLecture 10: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11)
Lecture 1: Eigenvectors and eigenvalues (Numerical Recipes, Chapter 11) The eigenvalue problem, Ax= λ x, occurs in many, many contexts: classical mechanics, quantum mechanics, optics 22 Eigenvectors and
More informationMath 19 Sample Final Exam Solutions
SSEA Summer 2017 Math 19 Sample Final Exam Solutions 1. [10 points] For what value of the constant k will the function: ) 1 x 2 sin, x < 0 fx) = x, x + k coskx), x 0 be continuous at x = 0? Explain your
More informationLinear Algebra (Review) Volker Tresp 2018
Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c
More information6, Quantum theory of Fluorescence
6, Quantum theory of Fluorescence 1 Quantum theory of Damping: Density operator 2 Quantum theory of Damping: Langevin equation 3 System-Reservoir Interaction 4 Resonance Fluorescence 5 Decoherence Ref:
More informationPotentially useful reading Sakurai and Napolitano, sections 1.5 (Rotation), Schumacher & Westmoreland chapter 2
Problem Set 2: Interferometers & Random Matrices Graduate Quantum I Physics 6572 James Sethna Due Friday Sept. 5 Last correction at August 28, 2014, 8:32 pm Potentially useful reading Sakurai and Napolitano,
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationModule 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers
Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More information1 Parametric Bessel Equation and Bessel-Fourier Series
1 Parametric Bessel Equation an Bessel-Fourier Series Recall the parametric Bessel equation of orer n: x 2 y + xy + (a 2 x 2 n 2 )y = (1.1) The general solution is given by y = J n (ax) +Y n (ax). If we
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationThe Ehrenfest Theorems
The Ehrenfest Theorems Robert Gilmore Classical Preliminaries A classical system with n egrees of freeom is escribe by n secon orer orinary ifferential equations on the configuration space (n inepenent
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationEC5555 Economics Masters Refresher Course in Mathematics September 2013
EC5555 Economics Masters Reresher Course in Mathematics September 3 Lecture 5 Unconstraine Optimization an Quaratic Forms Francesco Feri We consier the unconstraine optimization or the case o unctions
More informationNoether s theorem applied to classical electrodynamics
Noether s theorem applie to classical electroynamics Thomas B. Mieling Faculty of Physics, University of ienna Boltzmanngasse 5, 090 ienna, Austria (Date: November 8, 207) The consequences of gauge invariance
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationBasic Calculus Review
Basic Calculus Review Lorenzo Rosasco ISML Mod. 2 - Machine Learning Vector Spaces Functionals and Operators (Matrices) Vector Space A vector space is a set V with binary operations +: V V V and : R V
More informationFull-State Feedback Design for a Multi-Input System
Full-State Feedback Design for a Multi-Input System A. Introduction The open-loop system is described by the following state space model. x(t) = Ax(t)+Bu(t), y(t) =Cx(t)+Du(t) () 4 8.5 A =, B =.5.5, C
More informationCKM Matrix I. V ud V us V ub d. d s b
s = V u V us V u V c V cs V c s V t V ts V t flavour CKM matrix mass 18 parameters (9 complex elements) -5 relative quark phases (unoservale) -9 unitarity conitions - = 4 inepenent parameters 3 Euler angles
More informationThis property turns out to be a general property of eigenvectors of a symmetric A that correspond to distinct eigenvalues as we shall see later.
34 To obtain an eigenvector x 2 0 2 for l 2 = 0, define: B 2 A - l 2 I 2 = È 1, 1, 1 Î 1-0 È 1, 0, 0 Î 1 = È 1, 1, 1 Î 1. To transform B 2 into an upper triangular matrix, subtract the first row of B 2
More informationFrom Matrix to Tensor. Charles F. Van Loan
From Matrix to Tensor Charles F. Van Loan Department of Computer Science January 28, 2016 From Matrix to Tensor From Tensor To Matrix 1 / 68 What is a Tensor? Instead of just A(i, j) it s A(i, j, k) or
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationPHYSICS 234 HOMEWORK 2 SOLUTIONS. So the eigenvalues are 1, 2, 4. To find the eigenvector for ω = 1 we have b c.
PHYSICS 34 HOMEWORK SOUTIONS.8.. The matrix we have to diagonalize is Ω 3 ( 4 So the characteristic equation is ( ( ω(( ω(4 ω. So the eigenvalues are,, 4. To find the eigenvector for ω we have 3 a (3 b
More informationQuantum Physics II (8.05) Fall 2002 Assignment 3
Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai
More informationLinear Algebra using Dirac Notation: Pt. 2
Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018
More informationQuantum Mechanics C (130C) Winter 2014 Assignment 7
University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C) Winter 014 Assignment 7 Posted March 3, 014 Due 11am Thursday March 13, 014 This is the last problem
More information22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices
m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix
More information