Quantum Mechanics in a Nutshell

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Quantum Mechanics in a Nutshell"

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 ket-vectors 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 c-number), 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 co-vector or a bra-vectors), ψ (ψ 1, ψ n ) (8) when the column vector is known as a ket-vector 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  = ÎÂÎ = i  j i j (12) j=1 Eigenvectors i and their respective eigenvalues λ i of a linear operator  are defined by  i = λ i i (13) In a matrix representation the eigenvalues can be determined from a characteristic equation det (  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  = 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 c-number or complex numbers the operators are sometimes called the q-numbers 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,  a i ϕ i = a i  ϕ i, (17) where  is an n n matrix and ϕ i s are the n 1 column vectors The entries A ij of the matrix  can be obtained for a given set of basis vectors ϕ 1, ϕ 2 ϕ n from  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  on a Hilbert space there exist an adjoint (or Hermitian conjugate) operator defined as ( ψ,  ϕ ) ( ψ, ϕ ) (20)

7 CONTENTS 5 It follows that (AB) = B A (21) and ( ψ ) = ψ Â, (22) where ψ ψ (23) In a matrix representation an adjoint operator is defined as  (  ) T (24) where () is a complex conjugation and () T is a transpose operation Some useful definitions:  is a positive definite operator if ( ψ,  ψ ) is a positive real number for all ψ 0  is a self-adjoint (or Hermitian) operator if  = definite operator is Hermitian  Any positive  is a normal operator if  =   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 ket-vector) in the Hilbert space with unit length where ψ is a bra-vector 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 bra-vectors, 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 time-dependent Schrodinger equation i d ψ(t) = Ĥ ψ(t) (42) dt in contrast to the time-independent Schrodinger equation (40) Using (33), (34), (38) and (39) n can find the most general solution of the time-dependent 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 time-independent Schrodinger equation Note that each solution of the time-independent Schrodinger equation gives rise to a simple solution of the time-dependent 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  S e i Ĥt ψ S (0) = ψ H ÂH(t) ψ H = ÂH(t) (47)  H (t) e i Ĥt  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  S e i Ĥt + e i Ĥt  S d = i Ĥe i Ĥt  S e i Ĥt e i Ĥt  S i = i (Ĥ H (t) ÂH(t)Ĥ) i dt e Ĥt Ĥe i Ĥt = i where the commutator is defined as [Ĥ,  H (t) ] (50) [Â, ˆB]  ˆB ˆB (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 square-integrable 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 time-independent Schrodinger equation as ( with a ground state 2 ( ˆp 2 2m + kˆq2 2 2 ) 2m q + 2 kq2 Ψ(q) exp corresponding to the zero-point 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 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 information

Linear Algebra Review. Vectors

Linear 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 information

Quantum Mechanics and Representation Theory

Quantum 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 information

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality Inner Product Spaces and Orthogonality week 3-4 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 information

Quantum Physics II (8.05) Fall 2013 Assignment 4

Quantum 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 information

Operator methods in quantum mechanics

Operator 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 information

The Quantum Harmonic Oscillator Stephen Webb

The 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 information

13 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.

13 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 three-space, we write a vector in terms

More information

Quantum Time: Formalism and Applications

Quantum 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 information

Till now, almost all attention has been focussed on discussing the state of a quantum system.

Till 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 information

Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

Recall 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 n-dimensional 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 information

1 Lecture 3: Operators in Quantum Mechanics

1 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 information

Physics 221A Spring 2016 Notes 1 The Mathematical Formalism of Quantum Mechanics

Physics 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 information

Math 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 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 Self-Adjoint and Normal Operators

More information

Introduction to Matrix Algebra

Introduction 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 information

Factorization Theorems

Factorization 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 information

The Characteristic Polynomial

The 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 information

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Problem 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 information

1 Sets and Set Notation.

1 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 information

Time Ordered Perturbation Theory

Time 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 information

Linear Algebra: Vectors

Linear 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 information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture 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 information

α = u v. In other words, Orthogonal Projection

α = 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 information

x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

x + 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 information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. 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 information

Inner products on R n, and more

Inner 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 information

Quantum Computing. Robert Sizemore

Quantum 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 information

Chapter 7. Lyapunov Exponents. 7.1 Maps

Chapter 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 information

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear 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 information

Time dependence in quantum mechanics Notes on Quantum Mechanics

Time dependence in quantum mechanics Notes on Quantum Mechanics Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan

More information

PHYSICAL REVIEW A 79, 052110 2009. Multiple-time states and multiple-time measurements in quantum mechanics

PHYSICAL REVIEW A 79, 052110 2009. Multiple-time states and multiple-time measurements in quantum mechanics Multiple-time states and multiple-time 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 information

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The 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 orthogonal-diagonal-orthogonal type matrix decompositions Every

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

3 Orthogonal Vectors and Matrices

3 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 information

Lecture 18 - Clifford Algebras and Spin groups

Lecture 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 information

DATA ANALYSIS II. Matrix Algorithms

DATA 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 information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 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 information

MASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)

MASTER 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 non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced

More information

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear 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 information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall 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 information

Chapter 9 Unitary Groups and SU(N)

Chapter 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 information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 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 information

Similar matrices and Jordan form

Similar 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 information

Linear Algebra: Determinants, Inverses, Rank

Linear 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 information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES 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 information

LINEAR ALGEBRA W W L CHEN

LINEAR 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 information

State of Stress at Point

State 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 information

Is Quantum Mechanics Exact?

Is 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 information

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY 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 information

Generally Covariant Quantum Mechanics

Generally 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 information

The Limits of Adiabatic Quantum Computation

The 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 information

Principal Rotation Representations of Proper NxN Orthogonal Matrices

Principal 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 information

Math 312 Homework 1 Solutions

Math 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 information

Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics

Mehtap 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 information

THE KADISON-SINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT

THE KADISON-SINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT THE ADISON-SINGER 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 adison-singer

More information

1 Complex Numbers in Quantum Mechanics

1 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 information

Group Theory. 1 Cartan Subalgebra and the Roots. November 23, 2011. 1.1 Cartan Subalgebra. 1.2 Root system

Group 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 information

Irreducible Representations of SO(2) and SO(3)

Irreducible 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 information

Math 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 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 information

Fundamental Quantum Mechanics for Engineers

Fundamental 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 information

Rate of convergence towards Hartree dynamics

Rate 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 information

NOV - 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

NOV - 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 information

Orthogonal Projections

Orthogonal 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 information

Quantum Computing and Grover s Algorithm

Quantum 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 information

Matrices and Polynomials

Matrices 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 information

3. 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.

3. 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 information

Elements of Abstract Group Theory

Elements 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 information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Structure of the Root Spaces for Simple Lie Algebras

Structure 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 information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, 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 information

BILINEAR FORMS KEITH CONRAD

BILINEAR 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 information

An Introduction to Hartree-Fock Molecular Orbital Theory

An Introduction to Hartree-Fock Molecular Orbital Theory An Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction Hartree-Fock theory is fundamental

More information

Orthogonal Bases and the QR Algorithm

Orthogonal 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 information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Matrix Calculations: Applications of Eigenvalues and Eigenvectors; Inner Products

Matrix 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 information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE 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 information

Nonlinear Iterative Partial Least Squares Method

Nonlinear Iterative Partial Least Squares Method Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

More information

CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 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 information

Bits Superposition Quantum Parallelism

Bits Superposition Quantum Parallelism 7-Qubit 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 information

5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM

5.61 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 information

Space-Time Approach to Non-Relativistic Quantum Mechanics

Space-Time Approach to Non-Relativistic Quantum Mechanics R. P. Feynman, Rev. of Mod. Phys., 20, 367 1948 Space-Time Approach to Non-Relativistic Quantum Mechanics R.P. Feynman Cornell University, Ithaca, New York Reprinted in Quantum Electrodynamics, edited

More information

ENTANGLEMENT EXCHANGE AND BOHMIAN MECHANICS

ENTANGLEMENT 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 information

1 Symmetries of regular polyhedra

1 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 information

Section 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 5.3. 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 information

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

More information

Numerical Analysis Lecture Notes

Numerical 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 information

Grassmann Algebra in Game Development. Eric Lengyel, PhD Terathon Software

Grassmann 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 information

Using the Theory of Reals in. Analyzing Continuous and Hybrid Systems

Using 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 information

Lecture notes on linear algebra

Lecture 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 information

Prerequisite: High School Chemistry.

Prerequisite: 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 information

Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Physics 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 information

Introduction to Geometric Algebra Lecture II

Introduction 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 information

Simulation of quantum dynamics via classical collective behavior

Simulation of quantum dynamics via classical collective behavior arxiv:quant-ph/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 information

Typical Linear Equation Set and Corresponding Matrices

Typical 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 information

Linear Algebra I. Ronald van Luijk, 2012

Linear 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 information

Factor analysis. Angela Montanari

Factor 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 information

Assessment Plan for Learning Outcomes for BA/BS in Physics

Assessment 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 information

3. INNER PRODUCT SPACES

3. 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 information

Semi-Simple Lie Algebras and Their Representations

Semi-Simple Lie Algebras and Their Representations i Semi-Simple 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