Quantum Mechanics in a Nutshell


 Teresa Reynolds
 1 years ago
 Views:
Transcription
1 Quantum Mechanics in a Nutshell by Vitaly Vanchurin 1 November 13, If you have comments or error corrections, please send them to the LYX Documentation mailing list:
2 Contents 01 Vector Spaces 1 02 Inner Product 2 03 Outer Product 3 04 Linear Operators 4 05 Adjoint Operators 4 06 Classical Physics 6 07 Quantum Physics 7 08 Schrodinger Picture 8 09 Heisenberg Picture Harmonic Oscillator Ladder Operators 12 i
3 CONTENTS 1 01 Vector Spaces For our purpose the most relevant vector space is a finite (or countable) dimensional space of vectors with complex components represented in the column matrix notation as ψ ψ 1 ψ n where ψ i C and ψ C n or C These are the ketvectors By definition the vector spaces comes with: 1 addition operation ψ 1 ψ n + ϕ 1 ϕ n = ψ 1 + ϕ 1 ψ n + ϕ n (1), (2) 2 multiplication by a scalar, that is, a complex number (or cnumber), ψ 1 z ψ 1 z = (3) ψ n z ψ n 3 and a zero vector, not to be confused with a vacuum state 0 Spanning set in a vector space is a collection of vectors ϕ 1, ϕ 2 ϕ n such that an arbitrary vector can be written as their liner combination, (4) ψ = a i ϕ i (5)
4 CONTENTS 2 02 Inner Product An inner product (, ) is a function which takes two vectors ψ and ϕ to produce a single complex number denoted by ( ψ, ϕ ) with following properties: 1 (, ) is linear in the second argument 2 ( ψ, ϕ ) = ( ϕ, ψ ) 3 ( ψ, ψ ) 0 with equality only iff ψ = 0 For example the vector space C n has an inner product defined by ψ 1 ψ n, ϕ 1 ϕ n ψi ϕ i (6) The dual vector ψ is a linear function from a vector space to complex numbers defined using the inner product, ψ ( ϕ ) ψ ϕ ( ψ, ϕ ) (7) In matrix representation it is a raw vector (also known as a dual vector, a covector or a bravectors), ψ (ψ 1, ψ n ) (8) when the column vector is known as a ketvector In a quantum mechanics a finite dimensional vector space with an inner product is called a Hilbert space A number of definitions follow Two vectors ψ and ϕ are orthogonal if ψ ϕ = 0 Norm of a vector ψ is ψ ψ ψ is a unit vector if ψ ψ = 1 a set of vectors i is orthonormal if i i = 1 and i j = 0 for i j
5 CONTENTS 3 With an orthonormal basis an inner product can be written in a matrix representation as ϕ 1 ψ ϕ = ψ i i, ϕ j j = ψi ϕ j i j = ψi ϕ j = (ψ1,, ψn) j=1 j=1 (9) ϕ n 03 Outer Product Another convenient representation of linear operators is an outer product representation defined by, ψ ϕ ( ϕ ) ψ ϕ ϕ = ϕ ϕ ψ (10) For example an identity operator can be represented as a sum of outer product of an orthonormal basis, ie Î = i i (11) This is known as a completeness relation which can be used to obtain an outer product representations of operators, n Â = ÎÂÎ = i Â j i j (12) j=1 Eigenvectors i and their respective eigenvalues λ i of a linear operator Â are defined by Â i = λ i i (13) In a matrix representation the eigenvalues can be determined from a characteristic equation det ( Â λî) = 0 (14) Diagonalizable representation of an operator (also known as a orthonormal decomposition) is given by where the eigenvectors i form an orthonormal set Â = λ i i i (15)
6 CONTENTS 4 04 Linear Operators Linear operator on a vector space is a function which is linear in its inputs, ( n ) A a i ϕ i = a i A( ϕ i ) (16) (To emphasize the difference from cnumber or complex numbers the operators are sometimes called the qnumbers or quantum numbers) The two simplest operators are the identity operator Î, defined by Î ψ = ψ for all ψ zero operator ˆ0, defined by ˆ0 ψ = 0 for all ψ Similarly to vectors, operators A : C n C n can be represented in terms of matrices In matrix representation the linearity condition (16) can be rewritten as, Â a i ϕ i = a i Â ϕ i, (17) where Â is an n n matrix and ϕ i s are the n 1 column vectors The entries A ij of the matrix Â can be obtained for a given set of basis vectors ϕ 1, ϕ 2 ϕ n from Â ϕ j = A ij ϕ i (18) i An important example of operators on C 2 are the Pauli matrices, σ 0 σ 1 σ 2 σ 3 ( ) ( ) ( ) 0 i i 0 ( ) 1 0 (19) Adjoint Operators For every linear operator Â on a Hilbert space there exist an adjoint (or Hermitian conjugate) operator defined as ( ψ, Â ϕ ) (Â ψ, ϕ ) (20)
7 CONTENTS 5 It follows that (AB) = B A (21) and (Â ψ ) = ψ Â, (22) where ψ ψ (23) In a matrix representation an adjoint operator is defined as Â ( Â ) T (24) where () is a complex conjugation and () T is a transpose operation Some useful definitions: Â is a positive definite operator if ( ψ, Â ψ ) is a positive real number for all ψ 0 Â is a selfadjoint (or Hermitian) operator if Â = definite operator is Hermitian Â Any positive Â is a normal operator if ÂÂ = Â Â For example, any Hermitian operator is normal ˆP is a projection operator if ˆP = k i i, where i is an orthonormal basis and k n Û is a unitary operator if Û Û = Î Any pair of orthonormal basis ψ i and ϕ i can be used to define unitary operators, ie Û = n ψ i ϕ i Some useful results: Unitary operators preserve the inner product, (Û ψ, Û ϕ ) = ψ Û Û ϕ = ψ Î ϕ = ψ ϕ (25)
8 CONTENTS 6 06 Classical Physics Classical physics is based on the two postulates: 1 State space postulate: Any closed system is associated with even dimensional space called the phase space The state is described by a single point (or vector) in the phase space The state is specified by N position (usually denoted by q i s) and N momentum (usually denoted by p i s) coordinates: (q 1, q 2,, q N, p 1, p 2,, p N ) (26) What is the dimensionality of the phase space of a simple harmonic oscillator? What is the dimensionality of the phase space of a single particle in a box? How many real numbers needed to specify a state of the harmonic oscillator and how many needed to specify a single particle? 2 Evolution postulate: Evolution of any closed system is described by function on the phase space The function is called Hamiltonian and denoted by H(q 1, q 2,, q N, p 1, p 2,, p N ) and the evolution is described by the following equations ṗ i = H q i q i = H p i (27) For a simple harmonic oscillator the Hamiltonian is and the corresponding equations of motion are H(q, p) = p2 2m + kq2 2 (28) ṗ = kq q = p m (29) or equivalently q = ṗ m = kq m (30)
9 CONTENTS 7 07 Quantum Physics In contrast the quantum physics is based on three postulates We will first state the postulates and then introduce the necessary mathematical formalism that goes with it 1 State space postulate: Any closed system is associated with a Hilbert space The state of the system is described by a single point (or ketvector) in the Hilbert space with unit length where ψ is a bravector ψ (31) ψ ψ ψ ψ = 1 (32) 2 Evolution postulate: Evolution of any closed system is described by a unitary operator, ie where ψ(t 2 ) = Û(t 2 t 1 ) ψ(t 1 ) (33) Û(t 2 t 1 ) e i (t 2 t 1 )Ĥ (34) and Ĥ is a Hermitian operator known as Hamiltonian operator 3 Measurement postulate: A measurement is described by a collection of measurement operators { ˆM m } with probability of an outcome m given by p(m) = ψ ˆM m ˆM m ψ (35) where ˆM m is the Hermitian conjugate of ˆMm and the state after measurement ˆM m ψ p(m) (36) So to understand the quantum mechanics we have to understand what is a Hilbert space, ket and bravectors, unitary and Hermitian operators, etc The branch of mathematics which studies vector spaces and linear operators on those spaces is linear algebra
10 CONTENTS 8 08 Schrodinger Picture One can show that all normal operators have a diagonal representation, Â = λ i i i (37) Therefore all positive definite operators as well as Hermitian operators have diagonal representations Then one can define a function of Hermitian operators as n f(â) f(λ i ) i i (38) for an arbitrary function f This can be applied to Eq (33) since the Hamiltonian operator Ĥ must be Hermitian and thus has a spectral decomposition Ĥ = E n ψ n ψ n (39) where E n and ψ n are the eigenvalues and corresponding eigenstates of Hamiltonian operators, ie Ĥ ψ n = E n ψ n (40) The energy eigenstate ψ n with the lowest energy eigenvalue E n is called the ground state Moreover, by expanding both sides of Eq (33) we get ψ(t 1 ) + (t 2 t 1 ) d ( dt ψ(t 1) = 1 (t 2 t 1 ) i ) ψ(t 1 ) (41) Ĥ which is the famous timedependent Schrodinger equation i d ψ(t) = Ĥ ψ(t) (42) dt in contrast to the timeindependent Schrodinger equation (40) Using (33), (34), (38) and (39) n can find the most general solution of the timedependent Schrodinger equation ϕ (t) = n e ient [ ψ n ϕ (0)] ψ n (43) which is expressed as a linear sum over solutions of the timeindependent Schrodinger equation Note that each solution of the timeindependent Schrodinger equation gives rise to a simple solution of the timedependent Schrodinger equation ϕ (t) = e ient ψ n (44)
11 CONTENTS 9 09 Heisenberg Picture A framework where the state vectors evolves with time, but the operators remain constant is a Schrodinger picture There is also a Heisenberg picture where the operators change with time, and the state vectors remain constant Consider a time independent Hermitian operator ÂS in the Schrodinger picture then ÂS (t) = ψ S (t) ÂS ψ S (t) (45) The time evolution is described by a Schrodinger equation whose solution is From (45) and (46) we obtain where ψ S (t) = e i Ĥt ψ S (0) (46) ÂS (t) = ψ S (0) e i Ĥt Â S e i Ĥt ψ S (0) = ψ H ÂH(t) ψ H = ÂH(t) (47) Â H (t) e i Ĥt Â S e i Ĥt (48) ψ H ψ S (0) (49) It is easy to calculate the time evolution of operators in Heisenberg picture, dâh(t) dt = d dt e i Ĥt Â S e i Ĥt + e i Ĥt Â S d = i Ĥe i Ĥt Â S e i Ĥt e i Ĥt Â S i = i (ĤÂ H (t) ÂH(t)Ĥ) i dt e Ĥt Ĥe i Ĥt = i where the commutator is defined as [Ĥ, Â H (t) ] (50) [Â, ˆB] Â ˆB ˆBÂ (51) Expression (50) is known as the Heisenberg equations It reduces to the Hamiltonian equations of motion in classical mechanics with commutators replaced by Poisson brackets, i [, ] {, } q p p q (52)
12 CONTENTS 10 ie dp dt dq dt = {p, H} = H q = {q, H} = H p (53) (54) Moreover, the Poisson bracket between conjugate variables is replaced by a commutation relation {q, p} = 1 (55) i [ˆq, ˆp] = 1 (56) 010 Harmonic Oscillator The state of classical harmonic oscillator is described by position q(t) and momentum p(t) coordinate satisfying equations of motion (53) and (54) with Hamiltonian H = p2 2m + kq2 2 (57) The corresponding second order differential equation of motion is q + k q = 0 (58) m Thus, for classical harmonic oscillator the lowest energy solution is given by q(t) = q(t) = 0 To find a solution corresponding to the ground state we work in the position basis where q are the position eigenvectors, ie and define the wave function as ˆq q = q q (59) Ψ(q) q ψ (60) To quantize the harmonic oscillator we first defined a vector space of square integrable complex valued functions, ie f(q) 2 dq < (61)
13 CONTENTS 11 with an inner product given by, g f g(q) f(q)dq (62) It is easy to show that the space of squareintegrable complex valued functions is a Hilbert space Then we can replace the classical coordinated by quantum operators satisfying (56), eg q ˆq q (63) p ˆp i q (64) and rewrite the timeindependent Schrodinger equation as ( with a ground state 2 ( ˆp 2 2m + kˆq2 2 2 ) 2m q + 2 kq2 Ψ(q) exp corresponding to the zeropoint energy ( ) ψ = E ψ (65) Ψ(q) = EΨ(q) (66) ) km 2 q2 (67) ( ) ( ) 2 km E 0 = 2 = k 2m 2 2 m (68) where ω k/m is the angular frequency Then the solution of a timedependent Schrodinger equation is Ψ(q, t) = ( km π ) 1 4 exp ( i E 0 t ) km 2 q2 where the normalization is determined from Ψ(q) Ψ(q) = 1
14 CONTENTS 12 Then, for example, a set of measuring operators {M q = q q } describes probabilities (given by (35)) for detecting the particle in position q, p(q) = ψ M q ψ = ψ q q ψ = q ψ q ψ = Ψ(q) Ψ(q) ( ) km 1 ( ) 2 km = exp π 4 q2 This is the famous Born rule which is the additional third postulate of quantum mechanics which has no analog in classical physics It is a subject of ongoing debated whether the postulate can be derived from the first two (ie the state space and evolution postulates) 011 Ladder Operators It is convenient to define lowering â mω 2 ( ˆx + i ) mω ˆp and raising operator ( mω â ˆx i ) 2 mω ˆp which are useful for expressing other operators (69) (70) ˆx = (â + â ) (71) 2mω mω ˆp = i (â â ) (72) 2 ( Ĥ = â â + 1 ) ( ω = ââ 1 ) ω (73) 2 2 with following commutation relations, [â, â ] = 1 (74) [Ĥ, â ] = â ω (75) [Ĥ, â] = â ω (76)
15 CONTENTS 13 If we now denote the energy eigenstates ψ n with eigenvalues E n, ie then the commutation relations imply Ĥ ψ n = E n ψ n, (77) Ĥ â ψ n = (E n + ω)â ψ n (78) Ĥ â ψ n = (E n ω)â ψ n (79) This means that â ψ n and â ψ n are also eigenstates of the Hamiltonian with energies E n + ω and E n ω That is why operators â and â are called the lowering and rising operators between energy eigenstates However the eigenvalues cannot be arbitrarily small, ( E n = ψ n Ĥ ψ n = ψ n â â + 1 ) ω ψ n = (â ψ n ) â ψ n ω 1 2 ω (80) and thus, the ground state is defined by, â ψ 0 = 0 (81) in agreement with (67) Additional application of the lowering operator would not produce any new eigenstates, but the raising operator can be used to produce all of the excited states, ψ n = 1 (â ) n ψ0 (82) n! with energies, E n = ( n + 1 ) ω (83) 2
Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationQuantum Physics II (8.05) Fall 2013 Assignment 4
Quantum Physics II (8.05) Fall 2013 Assignment 4 Massachusetts Institute of Technology Physics Department Due October 4, 2013 September 27, 2013 3:00 pm Problem Set 4 1. Identitites for commutators (Based
More informationOperator methods in quantum mechanics
Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be
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 information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationQuantum Time: Formalism and Applications
Quantum Time: Formalism and Applications Submitted in partial fulfillment of honors requirements for the Department of Physics and Astronomy, Franklin and Marshall College, by Yuan Gao Professor Calvin
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 informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
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 informationPhysics 221A Spring 2016 Notes 1 The Mathematical Formalism of Quantum Mechanics
Copyright c 2016 by Robert G. Littlejohn Physics 221A Spring 2016 Notes 1 The Mathematical Formalism of Quantum Mechanics 1. Introduction The prerequisites for Physics 221A include a full year of undergraduate
More informationMath 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 SelfAdjoint and Normal Operators
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 informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationTime Ordered Perturbation Theory
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Quantization of the Free Electromagnetic Field We have so far quantized the free scalar field and the free Dirac field.
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
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 informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationx + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3
Math 24 FINAL EXAM (2/9/9  SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
More informationQuantum Computing. Robert Sizemore
Quantum Computing Robert Sizemore Outline Introduction: What is quantum computing? What use is quantum computing? Overview of Quantum Systems Dirac notation & wave functions Two level systems Classical
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationTime 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 13:22:3705:00 Copyright 2003 Dan
More informationPHYSICAL REVIEW A 79, 052110 2009. Multipletime states and multipletime measurements in quantum mechanics
Multipletime states and multipletime measurements in quantum mechanics Yakir Aharonov, 1, Sandu Popescu, 3,4 Jeff Tollaksen, and Lev Vaidman 1 1 Raymond and Beverly Sackler School of Physics and Astronomy,
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationLecture 18  Clifford Algebras and Spin groups
Lecture 18  Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for nonphysics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
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 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 informationIs Quantum Mechanics Exact?
Is Quantum Mechanics Exact? Anton Kapustin Simons Center for Geometry and Physics Stony Brook University This year Quantum Theory will celebrate its 90th birthday. Werner Heisenberg s paper Quantum theoretic
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationGenerally Covariant Quantum Mechanics
Chapter 15 Generally Covariant Quantum Mechanics by Myron W. Evans, Alpha Foundation s Institutute for Advance Study (AIAS). (emyrone@oal.com, www.aias.us, www.atomicprecision.com) Dedicated to the Late
More informationThe Limits of Adiabatic Quantum Computation
The Limits of Adiabatic Quantum Computation Alper Sarikaya June 11, 2009 Presentation of work given on: Thesis and Presentation approved by: Date: Contents Abstract ii 1 Introduction to Quantum Computation
More informationPrincipal Rotation Representations of Proper NxN Orthogonal Matrices
Principal Rotation Representations of Proper NxN Orthogonal Matrices Hanspeter Schaub Panagiotis siotras John L. Junkins Abstract hree and four parameter representations of x orthogonal matrices are extended
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationMehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics
INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree
More informationTHE KADISONSINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT
THE ADISONSINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT PETER G. CASAZZA, MATTHEW FICUS, JANET C. TREMAIN, ERIC WEBER Abstract. We will show that the famous, intractible 1959 adisonsinger
More information1 Complex Numbers in Quantum Mechanics
1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials,
More informationGroup Theory. 1 Cartan Subalgebra and the Roots. November 23, 2011. 1.1 Cartan Subalgebra. 1.2 Root system
Group Theory November 23, 2011 1 Cartan Subalgebra and the Roots 1.1 Cartan Subalgebra Let G be the Lie algebra, if h G it is called a subalgebra of G. Now we seek a basis in which [x, T a ] = ζ a T a
More informationIrreducible Representations of SO(2) and SO(3)
Chapter 8 Irreducible Representations of SO(2) and SO(3) The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard 1 Some of the most useful aspects of
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationFundamental Quantum Mechanics for Engineers
Fundamental Quantum Mechanics for Engineers Leon van Dommelen 5/5/07 Version 3.1 beta 3. ii Dedication To my parents iii iv Preface Why Another Book on Quantum Mechanics? This document was written because
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationNOV  30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane
Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationQuantum Computing and Grover s Algorithm
Quantum Computing and Grover s Algorithm Matthew Hayward January 14, 2015 1 Contents 1 Motivation for Study of Quantum Computing 3 1.1 A Killer App for Quantum Computing.............. 3 2 The Quantum Computer
More informationMatrices and Polynomials
APPENDIX 9 Matrices and Polynomials he Multiplication of Polynomials Let α(z) =α 0 +α 1 z+α 2 z 2 + α p z p and y(z) =y 0 +y 1 z+y 2 z 2 + y n z n be two polynomials of degrees p and n respectively. hen,
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationStructure of the Root Spaces for Simple Lie Algebras
Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationBILINEAR FORMS KEITH CONRAD
BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric
More informationAn Introduction to HartreeFock Molecular Orbital Theory
An Introduction to HartreeFock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction HartreeFock theory is fundamental
More informationOrthogonal Bases and the QR Algorithm
Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationMatrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products
Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products H. Geuvers Institute for Computing and Information Sciences Intelligent Systems Version: spring 2015 H. Geuvers Version:
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationBits Superposition Quantum Parallelism
7Qubit Quantum Computer Typical Ion Oscillations in a Trap Bits Qubits vs Each qubit can represent both a or at the same time! This phenomenon is known as Superposition. It leads to Quantum Parallelism
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 informationSpaceTime Approach to NonRelativistic Quantum Mechanics
R. P. Feynman, Rev. of Mod. Phys., 20, 367 1948 SpaceTime Approach to NonRelativistic Quantum Mechanics R.P. Feynman Cornell University, Ithaca, New York Reprinted in Quantum Electrodynamics, edited
More informationENTANGLEMENT EXCHANGE AND BOHMIAN MECHANICS
ENTANGLEMENT EXCHANGE AND BOHMIAN MECHANICS NICK HUGGETT TIZIANA VISTARINI PHILOSOPHY, UNIVERSITY OF ILLINOIS AT CHICAGO When two systems, of which we know the states by their respective representatives,
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded stepbystep through lowdimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
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 informationGrassmann Algebra in Game Development. Eric Lengyel, PhD Terathon Software
Grassmann Algebra in Game Development Eric Lengyel, PhD Terathon Software Math used in 3D programming Dot / cross products, scalar triple product Planes as 4D vectors Homogeneous coordinates Plücker coordinates
More informationUsing the Theory of Reals in. Analyzing Continuous and Hybrid Systems
Using the Theory of Reals in Analyzing Continuous and Hybrid Systems Ashish Tiwari Computer Science Laboratory (CSL) SRI International (SRI) Menlo Park, CA 94025 Email: ashish.tiwari@sri.com Ashish Tiwari
More informationLecture notes on linear algebra
Lecture notes on linear algebra David Lerner Department of Mathematics University of Kansas These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra
More informationPrerequisite: High School Chemistry.
ACT 101 Financial Accounting The course will provide the student with a fundamental understanding of accounting as a means for decision making by integrating preparation of financial information and written
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationIntroduction to Geometric Algebra Lecture II
Introduction to Geometric Algebra Lecture II Leandro A. F. Fernandes laffernandes@inf.ufrgs.br Manuel M. Oliveira oliveira@inf.ufrgs.br Visgraf  Summer School in Computer Graphics  2010 CG UFRGS Checkpoint
More informationSimulation of quantum dynamics via classical collective behavior
arxiv:quantph/0602155v1 17 Feb 2006 Simulation of quantum dynamics via classical collective behavior Yu.I.Ozhigov Moscow State University, Institute of physics and technology of RAS March 25, 2008 Abstract
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationFactor analysis. Angela Montanari
Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number
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 information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationSemiSimple Lie Algebras and Their Representations
i SemiSimple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book
More information