2.4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator

Size: px
Start display at page:

Download "2.4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator"

Transcription

1 .4. SPECTRAL DECOMPOSITION 63 Let P +, P, P 1,..., P p be the corresponding orthogonal complimentary system of projections, that is, P + + P + p P i = I. i=1 Then there exists a corresponding system of operators N 1,..., N p satisfying the equations N i = P i, N i P i = P i N i = N i, N i P j = P j N i = 0, if i j and the angles θ 1,... θ k such that π < θ i < π and O = P + P + p R i (θ i ) i=1 where R i (θ i ) = cos θ i P i + sin θ i N i. are the two-dimensional rotation operators in the planes corresponding to P i. Theorem.4.9 Every invertible operator A on a real vector space can be written in a unique way as a product A = OR of an orthogonal operator O and a symmetric positive operator R..4.8 Heisenberg Algebra, Fock Space and Harmonic Oscillator Heisenberg Algebra. The Heisenberg algebra is a 3-dimensional Lie algebra with generators X, Y, Z satisfying the commutation relations [X, Y] = Z, [X, Z] = 0, [X, Z] = 0. A representation of the Lie algebra A is a homomorphism ρ : A L(V) from the Lie algebra to the space of operators on a vector space V such that ρ([s, T]) = [ρ(s ), ρ(t)]. mathphyshass1.tex; September 4, 013; 9:58; p. 63

2 64 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES The Heisenberg algebra can be represented by matrices X =, Y = 0 0 1, Z = or by the differential operators C (R 3 ) C (R 3 ) defined by X = x 1 y z, Y = y + 1 x z, Z = z Properties of the Heisenberg algebra [X, Y n ] = nzy n 1 [Y, X n ] = nzx n 1 [X, exp(by)] = bz exp(by) [Y, exp(ax)] = az exp(ax) exp( by)x exp(by) = X + bz exp(ax)y exp( ax) = Y + az exp(ax) exp(by) = exp(abz) exp(by) exp(ax) exp( by) exp(ax) exp(by) = exp(abz) exp(ax) exp(ax) exp(by) exp( ax) = exp(abz) exp(by) exp(ax) exp(by) = exp(abz) exp(by) exp(ax) Another useful formula is X exp ( 1 ) Y = exp ( 1 ) Y (X ZY) Campbell-Hausdorff formula exp(ax + by) = exp ( ab ) Z exp(ax) exp(by) ( ) ab = exp Z exp(by) exp(ax) mathphyshass1.tex; September 4, 013; 9:58; p. 64

3 .4. SPECTRAL DECOMPOSITION 65 Heisenberg group. The Heisenberg group is a 3-dimensional Lie group with the generators X, Y, Z. An arbitrary element of the Heisenberg group is parametrized by canonical coordinates (a, b, c) as g(a, b, c) = exp(ax + by + cz) Obviously, and the inverse is defined by g(0, 0, 0) = I [g(a, b, c)] 1 = g ( a, b, c) The group multiplication law in the Heisenberg group takes the form ( g(a, b, c)g(a, b, c ) = g a + a, b + b, c + c + 1 ) (ab a b) Notice that ( g(a, b, c) = g 0, 0, c + ab ) g(0, b, 0)g(a, 0, 0) A representation of a Lie group G is a homomorphism ρ : G Aut (V) from the group G to the space of invertible operators on a vector space V such that for any g, h G ρ(gh) = ρ(g)ρ(h) and ρ(g 1 ) = [ρ(g)] 1, ρ(e) = I Representations of the Heisenberg group. The elements of the Heisenberg group could be represented by the uppertriangular matrices. Notice that X = Y = Z = XZ = YZ = 0 mathphyshass1.tex; September 4, 013; 9:58; p. 65

4 66 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES and or Therefore, 0 a c 0 0 b = XY = YX = Z 0 0 ab g(a, b, c) =, 1 a c + ab 0 1 b a c 0 1 b 0 0 1, 3 = 0. Another representation is defined by the action on functions in R 3. Notice that exp(ax) f (x, y, z) = f (x + a, y, z a ) y exp(by) f (x, y, z) = f (a, y + b, z + a ) x exp(cz) f (x, y, z) = f (x, y, z + c) Therefore, g(a, b, c) f (x, y, z) = f (x + a, y + b, z + c + b x a ) y Fock space. Let us define the operator N = YX It is easy to see that [N, Y] = ZY, [N, X] = ZX. Suppose that there exists a unit vector v 0 called the vacuum state such that Let us define a sequence of vectors Xv 0 = 0 v n = 1 n! Y n v 0 mathphyshass1.tex; September 4, 013; 9:58; p. 66

5 .4. SPECTRAL DECOMPOSITION 67 By using the properties of the Heisenberg algebra it is easy to show that Yv n = n + 1 v n+1, n 0, Xv n = n Zv n 1, n 1 Therefore Nv n = n Zv n, n 0 Let us define vectors w(b) = exp(by)v 0 = n=0 b n n! v n called the coherent states. Then by using the properties of the Heisenberg algebra we get Xw(b) = bzw(b) Now, suppose that and Y = X Z = I Then, the vectors v n are orthonormal (v n, v m ) = δ nm and are the eigenvectors of the self-adjoint operator N = X X with integer eigenvalues n 0. Then the space span {v n n 0} is called the Fock space and the operators X and X are called the annihilation and creation operators and the operator N is called the operator of the number of particles. mathphyshass1.tex; September 4, 013; 9:58; p. 67

6 68 CHAPTER. FINITE-DIMENSIONAL VECTOR SPACES Note, also that the coherent states are not orthonormal (w(a), w(b)) = eāb Finally, we compute the trace of the heat semigroup operator Tr exp( tx X) = e tn = n=0 1 1 e t Harmonic oscillator. Let D be an anti-self-adjoint operator and Q be a self-adjoint operator satisfying the commutation relations [D, Q] = I The harmonic oscillator is a quantum system with the (self-adjoint positive) Hamiltonian Then the operators H = 1 D + 1 Q X = 1 (D + Q), X = 1 ( D + Q). are the creation and annihilation operators. The operator of the number of particles is and, therefore, the Hamiltonian is N = X X = 1 D + 1 Q 1 H = N + 1 The eigenvalues of the Hamiltonian are λ n = n + 1 with the eigenvectors v n mathphyshass1.tex; September 4, 013; 9:58; p. 68

7 .4. SPECTRAL DECOMPOSITION 69 It is clear that the vectors ψ n (t) = e itλ n v n = exp satisfy the equation [ ( it n + 1 )] (i t H)ψ n = 0 which is called the Schrödinger equation. The vacuum state is determined from the equation (D + Q)v 0 = 0 1 n/ n! ( D + Q)n v 0 and has the form with ψ 0 satisfying v 0 = exp ( 1 ) Q ψ 0 Dψ 0 = 0. The heat trace (also called the partition function) for the harmonic oscillator is 1 Tr exp( th) = sinh(t/).4.9 Exercises 1. Find the eigenvalues of a projection operator.. Prove that the span of all eigenvectors corresponding to the eigenvalue λ of an operator A is a vector space. 3. Let E(λ) = Ker (A λi). Show that: a) if λ is not an eigenvalue of A, then E(λ) =, and b) if λ is an eigenvalue of A, then E(λ) is the eigenspace corresponding to the eigenvalue λ. 4. Show that the operator A λi is invertible if and only if λ is not an eigenvalue of the operator A. mathphyshass1.tex; September 4, 013; 9:58; p. 69

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2 Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition

More information

2.2. OPERATOR ALGEBRA 19. If S is a subset of E, then the set

2.2. OPERATOR ALGEBRA 19. If S is a subset of E, then the set 2.2. OPERATOR ALGEBRA 19 2.2 Operator Algebra 2.2.1 Algebra of Operators on a Vector Space A linear operator on a vector space E is a mapping L : E E satisfying the condition u, v E, a R, L(u + v) = L(u)

More information

Math 108b: Notes on the Spectral Theorem

Math 108b: Notes on the Spectral Theorem Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator

More information

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined

More information

Quantum Information & Quantum Computing

Quantum Information & Quantum Computing Math 478, Phys 478, CS4803, February 9, 006 1 Georgia Tech Math, Physics & Computing Math 478, Phys 478, CS4803 Quantum Information & Quantum Computing Problems Set 1 Due February 9, 006 Part I : 1. Read

More information

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For

More information

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Quantum mechanics in one hour

Quantum mechanics in one hour Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might

More information

1.4 Solvable Lie algebras

1.4 Solvable Lie algebras 1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)

More information

Introduction to the Mathematics of the XY -Spin Chain

Introduction to the Mathematics of the XY -Spin Chain Introduction to the Mathematics of the XY -Spin Chain Günter Stolz June 9, 2014 Abstract In the following we present an introduction to the mathematical theory of the XY spin chain. The importance of this

More information

Exercise Set 7.2. Skills

Exercise Set 7.2. Skills Orthogonally diagonalizable matrix Spectral decomposition (or eigenvalue decomposition) Schur decomposition Subdiagonal Upper Hessenburg form Upper Hessenburg decomposition Skills Be able to recognize

More information

Quantum Theory and Group Representations

Quantum Theory and Group Representations Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups

Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups Darren Funk-Neubauer Department of Mathematics and Physics Colorado State University - Pueblo Pueblo, Colorado, USA darren.funkneubauer@colostate-pueblo.edu

More information

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable

More information

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

Math 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler.

Math 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler. Math 453 Exam 3 Review The syllabus for Exam 3 is Chapter 6 (pages -2), Chapter 7 through page 37, and Chapter 8 through page 82 in Axler.. You should be sure to know precise definition of the terms we

More information

(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for

(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for Solutions to Homework #7 0. Prove that [S n, S n ] = A n for every n 2 (where A n is the alternating group). Solution: Since [f, g] = f 1 g 1 fg is an even permutation for all f, g S n and since A n is

More information

Problem Set # 8 Solutions

Problem Set # 8 Solutions Id: hw.tex,v 1.4 009/0/09 04:31:40 ike Exp 1 MIT.111/8.411/6.898/18.435 Quantum Information Science I Fall, 010 Sam Ocko November 15, 010 Problem Set # 8 Solutions 1. (a) The eigenvectors of S 1 S are

More information

MATH 53H - Solutions to Problem Set III

MATH 53H - Solutions to Problem Set III MATH 53H - Solutions to Problem Set III. Fix λ not an eigenvalue of L. Then det(λi L) 0 λi L is invertible. We have then p L+N (λ) = det(λi L N) = det(λi L) det(i (λi L) N) = = p L (λ) det(i (λi L) N)

More information

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

More information

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues

More information

Ph 219/CS 219. Exercises Due: Friday 20 October 2006

Ph 219/CS 219. Exercises Due: Friday 20 October 2006 1 Ph 219/CS 219 Exercises Due: Friday 20 October 2006 1.1 How far apart are two quantum states? Consider two quantum states described by density operators ρ and ρ in an N-dimensional Hilbert space, and

More information

2. Introduction to quantum mechanics

2. Introduction to quantum mechanics 2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian

More information

Second quantization: where quantization and particles come from?

Second quantization: where quantization and particles come from? 110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

Ph.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.

Ph.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified. PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system

More information

Density Matrices. Chapter Introduction

Density Matrices. Chapter Introduction Chapter 15 Density Matrices 15.1 Introduction Density matrices are employed in quantum mechanics to give a partial description of a quantum system, one from which certain details have been omitted. For

More information

1 Quantum field theory and Green s function

1 Quantum field theory and Green s function 1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory

More information

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4 Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary

More information

1. Matrix multiplication and Pauli Matrices: Pauli matrices are the 2 2 matrices. 1 0 i 0. 0 i

1. Matrix multiplication and Pauli Matrices: Pauli matrices are the 2 2 matrices. 1 0 i 0. 0 i Problems in basic linear algebra Science Academies Lecture Workshop at PSGRK College Coimbatore, June 22-24, 2016 Govind S. Krishnaswami, Chennai Mathematical Institute http://www.cmi.ac.in/~govind/teaching,

More information

Homework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9

Homework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9 Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year - Is

More information

Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1 Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

More information

Quantum Mechanics C (130C) Winter 2014 Assignment 7

Quantum 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 information

Physics 550. Problem Set 6: Kinematics and Dynamics

Physics 550. Problem Set 6: Kinematics and Dynamics Physics 550 Problem Set 6: Kinematics and Dynamics Name: Instructions / Notes / Suggestions: Each problem is worth five points. In order to receive credit, you must show your work. Circle your final answer.

More information

2 Quantization of the Electromagnetic Field

2 Quantization of the Electromagnetic Field 2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ

More information

5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:

5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis: 5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ

More information

Chapter 2 The Group U(1) and its Representations

Chapter 2 The Group U(1) and its Representations Chapter 2 The Group U(1) and its Representations The simplest example of a Lie group is the group of rotations of the plane, with elements parametrized by a single number, the angle of rotation θ. It is

More information

1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations

1 Planar rotations. Math Abstract Linear Algebra Fall 2011, section E1 Orthogonal matrices and rotations Math 46 - Abstract Linear Algebra Fall, section E Orthogonal matrices and rotations Planar rotations Definition: A planar rotation in R n is a linear map R: R n R n such that there is a plane P R n (through

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them. Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

More information

Diagonalization of Matrix

Diagonalization of Matrix of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that

More information

Tutorial 5 Clifford Algebra and so(n)

Tutorial 5 Clifford Algebra and so(n) Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called

More information

Linear Algebra 2 Spectral Notes

Linear Algebra 2 Spectral Notes Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex

More information

1 Homework 8 Solutions

1 Homework 8 Solutions 1 Homework 8 Solutions (1) Let G be a finite group acting on a finite set S. Let V be the vector space of functions f : S C and let ρ be the homomorphism G GL C (V ) such that (ρ(g)f)(s) = f(g 1 s) g G,

More information

Linear Algebra. Workbook

Linear Algebra. Workbook Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx

More information

Numerical Simulation of Spin Dynamics

Numerical Simulation of Spin Dynamics Numerical Simulation of Spin Dynamics Marie Kubinova MATH 789R: Advanced Numerical Linear Algebra Methods with Applications November 18, 2014 Introduction Discretization in time Computing the subpropagators

More information

Homework 11 Solutions. Math 110, Fall 2013.

Homework 11 Solutions. Math 110, Fall 2013. Homework 11 Solutions Math 110, Fall 2013 1 a) Suppose that T were self-adjoint Then, the Spectral Theorem tells us that there would exist an orthonormal basis of P 2 (R), (p 1, p 2, p 3 ), consisting

More information

MAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2)

MAT265 Mathematical Quantum Mechanics Brief Review of the Representations of SU(2) MAT65 Mathematical Quantum Mechanics Brief Review of the Representations of SU() (Notes for MAT80 taken by Shannon Starr, October 000) There are many references for representation theory in general, and

More information

Lecture 7: Positive Semidefinite Matrices

Lecture 7: Positive Semidefinite Matrices Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.

More information

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3]. Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

More information

Cartan s Criteria. Math 649, Dan Barbasch. February 26

Cartan s Criteria. Math 649, Dan Barbasch. February 26 Cartan s Criteria Math 649, 2013 Dan Barbasch February 26 Cartan s Criteria REFERENCES: Humphreys, I.2 and I.3. Definition The Cartan-Killing form of a Lie algebra is the bilinear form B(x, y) := Tr(ad

More information

Math 308 Practice Final Exam Page and vector y =

Math 308 Practice Final Exam Page and vector y = Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

Chapter 6 Inner product spaces

Chapter 6 Inner product spaces Chapter 6 Inner product spaces 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y

More information

Problem 1: Solving a linear equation

Problem 1: Solving a linear equation Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

The spectral action for Dirac operators with torsion

The spectral action for Dirac operators with torsion The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011 1 Torsion Geometry, Einstein-Cartan-Theory

More information

Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville Theory

Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville Theory Varied Factor Ordering in 2-D Quantum Gravity and Sturm-Liouville Theory Justin Rivera Wentworth Institute of Technology May 26, 2016 Justin Rivera (Wentworth Institute of Technology) Varied Factor Ordering

More information

Supplementary Notes on Linear Algebra

Supplementary Notes on Linear Algebra Supplementary Notes on Linear Algebra Mariusz Wodzicki May 3, 2015 1 Vector spaces 1.1 Coordinatization of a vector space 1.1.1 Given a basis B = {b 1,..., b n } in a vector space V, any vector v V can

More information

Linear Algebra using Dirac Notation: Pt. 2

Linear 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 information

Linear Algebra and Dirac Notation, Pt. 2

Linear Algebra and Dirac Notation, Pt. 2 Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14

More information

Intro to harmonic analysis on groups Risi Kondor

Intro to harmonic analysis on groups Risi Kondor Risi Kondor Any (sufficiently smooth) function f on the unit circle (equivalently, any 2π periodic f ) can be decomposed into a sum of sinusoidal waves f(x) = k= c n e ikx c n = 1 2π f(x) e ikx dx 2π 0

More information

Symmetric and anti symmetric matrices

Symmetric and anti symmetric matrices Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal

More information

Physics 215 Quantum Mechanics 1 Assignment 1

Physics 215 Quantum Mechanics 1 Assignment 1 Physics 5 Quantum Mechanics Assignment Logan A. Morrison January 9, 06 Problem Prove via the dual correspondence definition that the hermitian conjugate of α β is β α. By definition, the hermitian conjugate

More information

Linear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems.

Linear Algebra. Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems. Linear Algebra Chapter 8: Eigenvalues: Further Applications and Computations Section 8.2. Applications to Geometry Proofs of Theorems May 1, 2018 () Linear Algebra May 1, 2018 1 / 8 Table of contents 1

More information

Linear Algebra 2 Final Exam, December 7, 2015 SOLUTIONS. a + 2b = x a + 3b = y. This solves to a = 3x 2y, b = y x. Thus

Linear Algebra 2 Final Exam, December 7, 2015 SOLUTIONS. a + 2b = x a + 3b = y. This solves to a = 3x 2y, b = y x. Thus Linear Algebra 2 Final Exam, December 7, 2015 SOLUTIONS 1. (5.5 points) Let T : R 2 R 4 be a linear mapping satisfying T (1, 1) = ( 1, 0, 2, 3), T (2, 3) = (2, 3, 0, 0). Determine T (x, y) for (x, y) R

More information

Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:

Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis: Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein

More information

LIE ALGEBRAS: LECTURE 3 6 April 2010

LIE ALGEBRAS: LECTURE 3 6 April 2010 LIE ALGEBRAS: LECTURE 3 6 April 2010 CRYSTAL HOYT 1. Simple 3-dimensional Lie algebras Suppose L is a simple 3-dimensional Lie algebra over k, where k is algebraically closed. Then [L, L] = L, since otherwise

More information

One-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:

One-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation: One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's

More information

Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2. Review of Linear Algebra Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

More information

Quantum field theory and Green s function

Quantum field theory and Green s function 1 Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory

More information

The Spinor Representation

The Spinor Representation The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)

More information

Notation. For any Lie group G, we set G 0 to be the connected component of the identity.

Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence

More information

MATRIX LIE GROUPS AND LIE GROUPS

MATRIX LIE GROUPS AND LIE GROUPS MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either

More information

General Exam Part II, Fall 1998 Quantum Mechanics Solutions

General Exam Part II, Fall 1998 Quantum Mechanics Solutions General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V

More information

1 Last time: least-squares problems

1 Last time: least-squares problems MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

More information

Week Quadratic forms. Principal axes theorem. Text reference: this material corresponds to parts of sections 5.5, 8.2,

Week Quadratic forms. Principal axes theorem. Text reference: this material corresponds to parts of sections 5.5, 8.2, Math 051 W008 Margo Kondratieva Week 10-11 Quadratic forms Principal axes theorem Text reference: this material corresponds to parts of sections 55, 8, 83 89 Section 41 Motivation and introduction Consider

More information

Part IA. Vectors and Matrices. Year

Part IA. Vectors and Matrices. Year Part IA Vectors and Matrices Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2018 Paper 1, Section I 1C Vectors and Matrices For z, w C define the principal value of z w. State de Moivre s

More information

Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.

Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work. Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has

More information

Symmetric Spaces Toolkit

Symmetric Spaces Toolkit Symmetric Spaces Toolkit SFB/TR12 Langeoog, Nov. 1st 7th 2007 H. Sebert, S. Mandt Contents 1 Lie Groups and Lie Algebras 2 1.1 Matrix Lie Groups........................ 2 1.2 Lie Group Homomorphisms...................

More information

Mathematical foundations - linear algebra

Mathematical foundations - linear algebra Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar

More information

Definition (T -invariant subspace) Example. Example

Definition (T -invariant subspace) Example. Example Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

More information

Quantum Mechanics Solutions. λ i λ j v j v j v i v i.

Quantum Mechanics Solutions. λ i λ j v j v j v i v i. Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =

More information

Lecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1)

Lecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1) Lecture 19: Isometries, Positive operators, Polar and singular value decompositions; Unitary matrices and classical groups; Previews (1) Travis Schedler Thurs, Nov 18, 2010 (version: Wed, Nov 17, 2:15

More information

On the K-theory classification of topological states of matter

On the K-theory classification of topological states of matter On the K-theory classification of topological states of matter (1,2) (1) Department of Mathematics Mathematical Sciences Institute (2) Department of Theoretical Physics Research School of Physics and Engineering

More information

Lecture If two operators A, B commute then they have same set of eigenkets.

Lecture If two operators A, B commute then they have same set of eigenkets. Lecture 14 Matrix representing of Operators While representing operators in terms of matrices, we use the basis kets to compute the matrix elements of the operator as shown below < Φ 1 x Φ 1 >< Φ 1 x Φ

More information

Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed

Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like

More information

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal . Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P

More information

Physics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions

Physics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions Physics 37A Quantum Mechanics Fall 0 Midterm II - Solutions These are the solutions to the exam given to Lecture Problem [5 points] Consider a particle with mass m charge q in a simple harmonic oscillator

More information

Fermionic coherent states in infinite dimensions

Fermionic coherent states in infinite dimensions Fermionic coherent states in infinite dimensions Robert Oeckl Centro de Ciencias Matemáticas Universidad Nacional Autónoma de México Morelia, Mexico Coherent States and their Applications CIRM, Marseille,

More information

On the classification and modular extendability of E 0 -semigroups on factors Joint work with Daniel Markiewicz

On the classification and modular extendability of E 0 -semigroups on factors Joint work with Daniel Markiewicz On the classification and modular extendability of E 0 -semigroups on factors Joint work with Daniel Markiewicz Panchugopal Bikram Ben-Gurion University of the Nagev Beer Sheva, Israel pg.math@gmail.com

More information

MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

More information

Symmetries for fun and profit

Symmetries for fun and profit Symmetries for fun and profit Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 August 28, 2008 Sourendu Gupta (TIFR Graduate School) Symmetries for fun and profit QM I 1 / 20 Outline 1 The isotropic

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Postulates of Quantum Mechanics

Postulates of Quantum Mechanics EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 1: Stern-Gerlach experiment Postulates of Quantum Mechanics AStern-Gerlach(SG)deviceisabletoseparateparticlesaccordingtotheirspinalonga

More information

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson

Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Name: TA Name and section: NO CALCULATORS, SHOW ALL WORK, NO OTHER PAPERS ON DESK. There is very little actual work to be done on this exam if

More information

Linear Algebra: Matrix Eigenvalue Problems

Linear Algebra: Matrix Eigenvalue Problems CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

More information

Linear Algebra Review. Vectors

Linear 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 information