Mathematical Introduction

Transcription

1 Chapter 1 Mathematical Introduction HW #1: 164, 165, 166, 181, 182, 183, 1811, 1812, Linear Vector Spaces: Basics 111 Field A collection F of elements a,b etc (also called numbers or scalars) with two binary operations ( : F F F) and(+:f F F) is called a field if: associativity: (a + b)+c = a +(b + c) and(a b) c = a (b c) commutativity: a + b = b + a and a b = b a distributivity: (a + b) c =(b c)+(a c) existence of identity elements: a +=a and a 1=a existence of inverses: a +( a) =anda (a 1 ) = 1 (where the latter for all a ) The most familiar field is a field of real numbers R, but in this course we will mostly be looking at a field complex numbers C There are also other fields such a p-adic numbers that proved to be useful in theoretical physics 112 Vector space A collection V of elements ψ, ϕ, etc (also called vectors or ket vectors) wth one binary operation ( : V V V) and infinity many scalar multiplication operations (a : V V) is called a linear vector space if: 3

2 CHAPTER 1 MATHEMATICAL INTRODUCTION 4 associativity: a (b ψ ) =(a b) ψ and ψ ( ϕ χ ) =( ψ ϕ ) χ commutativity: ψ ϕ = ϕ ψ distributivity: (a + b)( ψ ϕ ) = a ψ b ψ a ϕ b ϕ existence of null vector: + ψ = ψ existence of inverse: ψ + ψ = For our purpose the most relevant vector space is a finite (or countable) dimensional space of vectors with complex components often represented in the column matrix notation as ψ ψ 1 ψ n where ψ i C and ψ C n or C Thesearetheket-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 (11), (12) 2 multiplication by a scalar, that is, a complex number (or c-number), ψ 1 zψ 1 z = (13) ψ n zψ n 3 and a zero vector, (not to be confused with a vacuum state ) (14)

3 CHAPTER 1 MATHEMATICAL INTRODUCTION 5 In quantum mechanics vectors represent states of the system in question These vectors evolve from one state to another according to some predetermined rule (ie Schrodinger equation) Exercise: Prove that is unique ψ = ψ = ψ ψ is unique 113 Dimensionality A set of vectors i is linearly independent if n a i i = (15) implies a i = for all i Otherwise the vectors i are linearly dependent The vector space has dimension n if it can accommodate a maximum of n linearly independent vectors (also called spanning set) It will be denoted by V n (R) if the field is real and V n (C) if the field is complex What is the dimensionality of the following vector spaces? R n or a collection of n real numbers M n m or a collection of nby m matrices L 2 or square-integrable functions G or a space of generalized functions (ie distributions) Exercise Prove that any vector in n dimensional space can be written as a linear combination of n linearly independent vectors 1, 2 n Note that in quantum mechanics the dimensionality of relevant vector space is usually (but not always) very large For example even a single Hydrogen atom requires an infinite dimensional vector spaces to describe its state

4 CHAPTER 1 MATHEMATICAL INTRODUCTION Basis vectors A set of n linearly independent vectors in an n dimensional space is called a basis and vectors are called basis vectors Then an arbitrary vector can be expanded as ψ = n ψ i i (16) where v i are called the components of vector ψ Exercise Prove that the expansion 16 is unique Note that multiplication by a scalar and addition of vectors written in terms of components follows from a definition of vector spaces: ( n ) n a ψ = a ψ i i = (a ψ i ) i (17) ψ ϕ = n ψ i i 12 Dual space 121 Hilbert space n ϕ i i = n (ψ i + ϕ i ) i (18) The vector space of quantum mechanical systems is called Hilbert space and it has one more additional structure: the inner product Inner product (or dot product or scalar product) is a map which satisfies the following properties: (, ) :V V C (19) Conjugation: ( ψ, ϕ ) =( ϕ, ψ ) where * denotes complex conjugation Linearity (with respect to second argument): ( ψ,a ϕ b χ ) = a ( ψ, ϕ )+ b ( ϕ, χ ) Positivity: ( ψ, ψ ) Exercise Show that ( ψ, ϕ ) = i (ψi ϕ j )( i, j ) (11) j The inner product allows us to define:

5 CHAPTER 1 MATHEMATICAL INTRODUCTION 7 Orthogonality of two vectors if their inner product vanishes, ie ( ψ, ϕ ) = (111) Norm of a vector as ψ = ( ψ, ψ ) (112) Orthonormal basis vectors such that ( i, j ) =δ ij (113) where δ ij = { if i j 1 if i = j (114) is the Kronecker delta symbol Then any vector can be written in an orthonormal basis i as ψ = i ( i, ψ ) i (115) 122 Bra vectors Given a space of ket vectors V and a particular ket vector ψ we can define amap ψ : V C (116) given by the following expression ψ ( ϕ ) ( ψ, ϕ ) (117) This map is what we call a bra vector ψ To simplify notations it is often denoted by ψ ϕ ( ψ, ϕ ) (118) (with no extra parenthesis) but it is important to understand that bra vector is a function ψ ( ) andket-vector ϕ is an argument of that function Exercise Show that the space of bra functions ψ ( ) also forms a vector space and for that reason it is usually called a bra vector Using the bra vector notation decomposition (115) can be written as ψ = i i i ψ (119)

6 CHAPTER 1 MATHEMATICAL INTRODUCTION 8 which can be thought of as an insertion of identity operator (also known as completeness relation) Î = i i (12) i where the basis vector are usually assumed to be orthonormal (also known as orthogonality relation) i j = δ ij 123 Gram-Schmidt theorem There is an important theorem (known as Gram-Schmidt theorem) which states that given a linearly independent basis we can always form linear combinations of the basis vectors to obtain an orthonormal basis The prove is constructive and we leave it as a homework to work out the details of the prove There are, however, two very important inequalities that we shall prove in class 124 Triangle inequality Triangle inequality states that ψ + ϕ ψ + ϕ (121) for an arbitrary pair of ket vectors ψ and ϕ To prove it we decompose both vectors in orthonormal basis then ψ + ϕ 2 = i (ψ i + ϕ i ) (ψ i + ϕ i ) = i ψ i ψ i +2 i (ϕ i ψ i + ψ i ϕ i )+ i ϕ i ϕ i = ψ 2 + ϕ 2 +2R ϕ ψ ψ 2 + ϕ 2 +2 ϕ ψ ψ 2 + ϕ 2 +2 ϕ ψ = ( ψ + ϕ ) 2 (122) but since both ψ + ϕ and ψ + ϕ are positive definite we have (121)

7 CHAPTER 1 MATHEMATICAL INTRODUCTION Schwartz inequalities Schwartz inequality states that ψ ϕ ψ ϕ (123) for an arbitrary pair of ket vectors ψ and ϕ To prove it we consider a vector χ = ψ ϕ ψ 2 ϕ (124) ϕ and apply the positivity axiom of dot product we get or χ χ = ψ ψ ϕ ϕ ψ 2 ϕ ψ ϕ ϕ 2 ϕ = ψ ψ ψ ϕ ϕ 2 ϕ ψ ϕ ψ ψ ϕ ϕ ψ 2 ψ ϕ + ϕ ϕ 2 ϕ 2 ϕ ϕ = ψ ψ ψ ϕ ϕ ψ ϕ 2 and (123) follows = ψ 2 ψ ϕ ϕ 2 (125) 13 Linear operators 131 Definition ψ 2 ψ ϕ ϕ 2 (126) So far we have only described states of the system as vectors in Hilbert space To describe evolution (and also observations) we need to consider linear operators  (often denoted with a hat) which can be thought of as a map  : V V (127) satisfying linearly property  (a ψ + b ϕ ) =aâ( ψ )+Â(b ϕ ) (128) Parenthesis are often emitted and the right hand side of (128)is written as aâ ψ + bâ ϕ (129)

8 CHAPTER 1 MATHEMATICAL INTRODUCTION 1 (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, definedbyî ψ = ψ for all ψ zero operator ˆ, defined by ˆ ψ = for all ψ An important feature of linear operators is that their action on basis vectors uniquely determines how they act on other vectors,  ψ = i ψ i  i 132 Adjoint operators By acting on a ket vector the corresponding bra vectors (or bra maps) would also transform This allows us to define adjoint operators that act on bra vectors according to ( ψ,  ϕ ) ( ψ, ϕ ) (13) It follows that ) ( ˆB = ˆB  (131) and ( ψ ) = ψ Â, (132) where ψ ψ (133) 133 Matrix representation In the so-called matrix representation: ψ ket vectors are column N 1 matrices (or vectors), ψ bra vectors are row 1 N matrices (or vectors)  operators are square N N matrices and adjoint operators can be defined as ) T  ( (134) where () is a complex conjugation and () T is a transpose operation

9 CHAPTER 1 MATHEMATICAL INTRODUCTION 11 Then expressions like ψ câ ϕ + b ϕ χ a χ ˆB ϕ (135) can be understood through matrix additions, matrix multiplications, and matrix multiplications by scalars More precisely a matrix representation can always be obtained by contracting the operator with bra and ket basis vectors, ie ψ i = i ψ (136) ψ i = ψ i (137) 134 Useful operators A ij = i  j (138) Here we summarize some useful definitions of operators which will appear when we start describe quantum systems  is a positive definite operator if ( ψ,  ψ ) is a positive real number for all ψ  is a Hermitian (or self-adjoint) operator if  = definite operator is Hermitian  Any positive  is a anti-hermitian operator if  =  Every operator can be decomposed into a sum of Hermitian and anti-hermitian operators  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 Exercise Show that the inner product is conserved under actions of unitary operators: (Û ψ, Û ϕ ) = ψ Û Û ϕ = ψ Î ϕ = ψ ϕ (139) Conservation of inner products is related to conservation of probabilities and is a fundamental property of nature at all scales For some time it was

10 CHAPTER 1 MATHEMATICAL INTRODUCTION 12 thought that evolution of black-holes is not unitary which gave rise to the so-called information paradox Note that out of all of the operators mentioned above the most essential for understanding quantum mechanics are: Hermitian operators (describes observables) Projection operators (describes measurements) Unitary operators (describes evolutions) 135 Eigenvalues and eigenvectors Eigenvectors i and their respective eigenvalues λ i of a linear operator  are defined by  i = λ i i (14) In a matrix representation the eigenvalues can be determined from acharac- teristic equation ) det ( λ Î = (141) Diagonalizable representation of an operator (also known as a orthonormal decomposition) is given by  = n λ i i i (142) where the eigenvectors i form an orthonormal set 136 Decompositions of operators Spectral decomposition Any normal operator M on vector space V is diagonal with respect to some orthonormal basis i s for V, M = i λ i i i Conversely, any diagonalizable operator is normal For Hermitian operators the eigenvalues are real Note that the spectral decomposition can be used to defined functions of operators, f(m) f(λ i ) i i i

11 CHAPTER 1 MATHEMATICAL INTRODUCTION 13 Polar decomposition An arbitrary linear operator can be decomposed into product of unitary operator U and positive operators J and K such that A = UJ = KU where J A A K AA Singular value decomposition For any square matrix A there are exit unitary matrices U and V, and a diagonal matrix D such that A = UDV The non-negative diagonal elements of D are called the singular values of A 14 Active and Passive transformations A framework where the state vectors evolves with time, but the operators remain constant is a Schrodinger picture The Schrodinger picture gives rise to active transformations of vectors (vectors are transforming) There is also a Heisenberg picture where the operators change with time, and the state vectors remain constant The Heisenberg picture gives rise to passive transformations of vectors (vectors are not really transforming) Consider a time independent Hermitian operator ÂS in the Schrodinger picture then expectation values of observables (ie Hermitian operators ) is defined by ÂS (t) ψ S (t) ÂS ψ S (t) (143) The time evolution will be described by a Schrodinger equation, but for now all we need to know is that there exist a unitary operator such that ψ S (t) = Û(t) ψ S() (144) The operator is sometimes called propagator (or evolution operators) because it propagates (or evolves) the state vector from time to time t) From (143) and(144) we obtain ÂS (t) = ψ S () Û (t)âsû(t) ψ S() = ψ H ÂH(t) ψ H = ÂH(t) (145)

12 CHAPTER 1 MATHEMATICAL INTRODUCTION 14 where  H (t) 15 Infinite dimensions Û (t)âsû(t) (146) ψ H ψ S () (147) So far we have mostly worked with abstract Dirac notations where there is no need to specify the dimensionality of the Hilbert space In matrix representation the dimensionality of bra and ket vectors is usually finite which would not capture all of the systems that we would like to study In what follows we will generalize the concept of matrix representation of a wave function representation and this will allows us to capture a lot more It is worth mentioning that the generalization must not stop here and one can imagine generalizing wave functions to wave functionals, but this is beyond the scope of this course Consider all possible (complex-valued) functions ψ(x) on interval from to L If we are to discretize the interval (with N = L/ε lattice points), then these functions can be represented by a column vector (ie ket-vector) with components ψ i ψ(iε) (148) where only values of the function ψ(x) at lattice sites x = {ε, 2ε,, Nε} are important So far the situation is not any different from matrix representation and we might want to proceed exactly as before In particular the inner product could be given by ψ ϕ = N ψi ϕ i = N ψ (iε) ϕ (iε) (149) This is however not very satisfactory since in the limit ε the summation would diverge An alternative definition of the inner product is ψ ϕ = N ψi ϕ i ε = N ψ (iε) ϕ (iε) ε (15) and then in a continuum ε limit the inner product is nothing but integration ψ ϕ = lim ε N ψ (iε) ϕ (iε) ε = ψ (x)ϕ(x)dx (151)

13 CHAPTER 1 MATHEMATICAL INTRODUCTION 15 An important thing to note here is that not all of the complex-valued functions might be square integrable, but we shall restrict ourselves only to those functions that are and will moreover impose a normalization condition ψ (x)ψ(x)dx =1 (152) All of the functions which satisfy (152)we will call wave-functions and will think of them as a representation for abstract ket vectors and bra vectors ψ ψ(x) (153) ψ ψ (x) (154) The complication arises when we start constructing basis vectors y Turns out that the basis vectors do not represent physical states and are called generalized vectors y δ(x y) y δ(x y) (155) These vector do not to satisfy the normality condition (152), instead the normalization is given by x y = δ(x z)δ(y z)dz = δ(x y) (156) Nevertheless the generalized basis vectors are very useful For example they can be used to construct wave-functions from ket vectors and vise versa ψ = ψ(x) = x ψ (157) ψ(x) x dx (158) There are not many operators on the space of wave-functions that we will need in this course In fact we will only use position operators ŷ : ψ(x) [xψ(x)] x=y (159) momentum operator ˆp y : ψ(x) [ i dψ(x) ] dx x=y (16)

14 CHAPTER 1 MATHEMATICAL INTRODUCTION 16 and combinations of these operators These operators can be represented in coordinate basis ˆx = ˆp = Then it is easy to see that and x x x dx (161) z ˆx ψ = z ˆp ψ = = = = [ = i d δ(x y) x y dxdy (162) dx xδ(z x)ψ(x)dx = zψ(z) (163) i d δ(x y)δ(z x)ψ(y)dxdy dx i d δ(x y)δ(z x)ψ(y)dxdy dy i d δ(z y)ψ(y)dy dy i δ(z y) d dy ψ(y)dy = ] (164) i d dy ψ(y) y=z It is also straightforward to show that the position and momentum operators satisfy the commutation relation [ˆx, ˆp] ˆxˆp ˆpˆx = = = = i = i i i ( d i d δ(x y)δ(x z)z z y dx dx ( ) d d δ(x y)x δ(x y)y x y dxdy dx dx ) ( δ(x y) d dx x x x dx x y dxdy ) δ(x y)δ(y z)z x z dxdydz (165)

15 CHAPTER 1 MATHEMATICAL INTRODUCTION 17 then One can also define momentum basis as ˆp p = p p (166) i d δ(x y) x y p dxdy = p p dx i d δ(x y) y p x dxdy = p p dy i δ(x y) d y p x dxdy = p p dy i d y p y dy = p p dy i d y p dy = p y p i d dy ψ p(y) = pψ p (y) (167) Clearly the position space wave function which satisfies above equation is ( ) i ψ p (y) exp py (168) where p = πn (169) for some integer n so that the wave function vanishes at the boundaries y = and y = L In the limit of infinite box all possible values of p are allowed and we can use the momentum basis to represent states and operators exactly as we used position basis For example in momentum basis ˆp = ˆx = p p p dp (17) i d δ(p r) p r dpdr (171) dp Moreover we can define momentum basis wave functions as ψ(p) = p ψ (172)