Dynamics a set of notes on theoretical physical chemistry 2003 steen, range york

Dynamics: A Set of Notes on Theoretical Physical Chemistry Jaclyn Steen, Kevin Range and Darrin M York December 5, 2003 Contents Vector Calculus 1.1 Properties of vectors and vector space 1.2 Fundamental operations involving vectors Linear Algebra 2.1 Matrices, Vectors and Scalars 2.2 Matrix Operations 2.3 Transpose of a Matrix 2.4 Unit Matrix 2.5 Trace of a (Square) Matrix 2.5.1 Inverse of a (Square) Matrix 2.6 More on Trace 2.7 More on [A, B] 2.8 Determinants 2.8.1 Laplacian expansion 2.8.2 Applications of Determinants 2.9 Generalized Green’s Theorem 2.10 Orthogonal Matrices 2.11 Symmetric/Antisymmetric Matrices 2.12 Similarity Transformation 2.13 Hermitian (self-adjoint) Matrices 2.14 Unitary Matrix 2.15 Comments about Hermitian Matrices and Unitary Tranformations 2.16 More on Hermitian Matrices 2.17 Eigenvectors and Eigenvalues 2.18 Anti-Hermitian Matrices 2.19 Functions of Matrices 2.20 Normal Marices 2.21 Matrix 2.21.1 Real Symmetric 2.21.2 Hermitian 2.21.3 Normal 2.21.4 Orthogonal 2.21.5 Unitary 6 11 11 11 12 12 13 13 13 13 14 14 15 16 17 17 18 18 18 18 18 19 19 19 21 21 21 21 21 21 21 CONTENTS CONTENTS Calculus of Variations 3.1 Functions and Functionals 3.2 Functional Derivatives 3.3 Variational Notation 3.4 Functional Derivatives: Elaboration 3.4.1 Algebraic Manipulations of Functional Derivatives 3.4.2 Generalization to Functionals of Higher Dimension 3.4.3 Higher Order Functional Variations and Derivatives 3.4.4 Integral Taylor series expansions 3.4.5 The chain relations for functional derivatives 3.4.6 Functional inverses 3.5 Homogeneity and convexity 3.5.1 Homogeneity properties of functions and functionals 3.5.2 Convexity properties of functions and functionals 3.6 Lagrange Multipliers 3.7 Problems 3.7.1 Problem 3.7.2 Problem 3.7.3 Problem 3.7.3.1 Part A 3.7.3.2 Part B 3.7.3.3 Part C 3.7.3.4 Part D 3.7.3.5 Part E 3.7.4 Problem 3.7.4.1 Part F 3.7.4.2 Part G 3.7.4.3 Part H 3.7.4.4 Part I 3.7.4.5 Part J 3.7.4.6 Part K 3.7.4.7 Part L 3.7.4.8 Part M 22 22 23 24 25 25 26 27 28 29 30 30 31 32 34 35 35 35 35 35 36 36 36 36 36 36 36 37 37 37 38 38 38 40 40 40 41 46 46 49 50 Variational Principles 5.1 Hamilton’s Principle 5.2 Comments about Hamilton’s Principle 5.3 Conservation Theorems and Symmetry 54 55 56 60 Classical Mechanics 4.1 Mechanics of a system of particles 4.1.1 Newton’s laws 4.1.2 Fundamental definitions 4.2 Constraints 4.3 D’Alembert’s principle 4.4 Velocity-dependent potentials 4.5 Frictional forces CONTENTS 61 61 61 61 61 62 63 64 67 68 68 71 Scattering 7.1 Introduction 7.2 Rutherford Scattering 7.2.1 Rutherford Scattering Cross Section 7.2.2 Rutherford Scattering in the Laboratory Frame 7.3 Examples 73 73 76 76 76 77 Collisions 8.1 Elastic Collisions 78 79 Oscillations 9.1 Euler Angles of Rotation 9.2 Oscillations 9.3 General Solution of Harmonic Oscillator Equation 9.3.1 1-Dimension 9.3.2 Many-Dimension 9.4 Forced Vibrations 9.5 Damped Oscillations Central Potential and More 6.1 Galilean Transformation 6.2 Kinetic Energy 6.3 Motion in 1-Dimension 6.3.1 Cartesian Coordinates 6.3.2 Generalized Coordinates 6.4 Classical Viral Theorem 6.5 Central Force Problem 6.6 Conditions for Closed Orbits 6.7 Bertrand’s Theorem 6.8 The Kepler Problem 6.9 The Laplace-Runge-Lenz Vector CONTENTS 10 Fourier Transforms 10.1 Fourier Integral Theorem 10.2 Theorems of Fourier Transforms 10.3 Derivative Theorem Proof 10.4 Convolution Theorem Proof 10.5 Parseval’s Theorem Proof 82 82 82 85 85 86 87 88 90 90 91 91 92 93 11 Ewald Sums 11.1 Rate of Change of a Vector 11.2 Rigid Body Equations of Motion 11.3 Principal Axis Transformation 11.4 Solving Rigid Body Problems 11.5 Euler’s equations of motion 11.6 Torque-Free Motion of a Rigid Body 11.7 Precession in a Magnetic Field 11.8 Derivation of the Ewald Sum 11.9 Coulomb integrals between Gaussians 95 95 95 97 97 98 98 99 100 100 CONTENTS 11.10Fourier Transforms 11.11Linear-scaling Electrostatics 11.12Green’s Function Expansion 11.13Discrete FT on a Regular Grid 11.14FFT 11.15Fast Fourier Poisson CONTENTS 12 Dielectric 12.1 Continuum Dielectric Models 12.2 Gauss’ Law I 12.3 Gauss’ Law II 12.4 Variational Principles of Electrostatics 12.5 Electrostatics - Recap 12.6 Dielectrics 101 103 103 104 104 105 106 106 108 109 109 110 112 13 Exapansions 13.1 Schwarz inequality 13.2 Triangle inequality 13.3 Schmidt Orthogonalization 13.4 Expansions of Functions 13.5 Fourier Series 13.6 Convergence Theorem for Fourier Series 13.7 Fourier series for different intervals 13.8 Complex Form of the Fourier Series 13.9 Uniform Convergence of Fourier Series 13.10Differentiation of Fourier Series 13.11Integration of Fourier Series 13.12Fourier Integral Representation 13.13M-Test for Uniform Convergence 13.14Fourier Integral Theorem 13.15Examples of the Fourier Integral Theorem 13.16Parseval’s Theorem for Fourier Transforms 13.17Convolution Theorem for Fourier Transforms 13.18Fourier Sine and Cosine Transforms and Representations 115 116 116 117 118 124 124 128 130 131 132 132 135 136 136 139 141 142 143 Chapter Vector Calculus These are summary notes on vector analysis and vector calculus The purpose is to serve as a review Although the discussion here can be generalized to differential forms and the introduction to tensors, transformations and linear algebra, an in depth discussion is deferred to later chapters, and to further reading.1, 2, 3, 4, For the purposes of this review, it is assumed that vectors are real and represented in a 3-dimensional Carteˆ, z ˆ), unless otherwise stated Sometimes the generalized coordinate notation x1 , x2 , x3 will be sian basis (ˆ x, y used generically to refer to x, y, z Cartesian components, respectively, in order to allow more concise formulas to be written using using i, j, k indexes and cyclic permutations If a sum appears without specification of the index bounds, assume summation is over the entire range of the index 1.1 Properties of vectors and vector space A vector is an entity that exists in a vector space In order to take for (in terms of numerical values for it’s components) a vector must be associated with a basis that spans the vector space In 3-D space, for example, a ˆ, z ˆ) This is an example of an orthonormal basis in that each component Cartesian basis can be defined (ˆ x, y ˆ·x ˆ=y ˆ·y ˆ=z ˆ·z ˆ = and orthogonal to the other basis vectors x ˆ·y ˆ=y ˆ·z ˆ= basis vector is normalized x ˆ·x ˆ = More generally, a basis (not necessarily the Cartesian basis, and not necessarily an orthonormal basis) is z denoted (e1 , e2 , e3 If the basis is normalized, this fact can be indicated by the “hat” symbol, and thus designated ˆ2 , e ˆ3 (ˆ e1 , e Here the properties of vectors and the vector space in which they reside are summarized Although the present chapter focuses on vectors in a 3-dimensional (3-D) space, many of the properties outlined here are more general, as will be seen later Nonetheless, in chemistry and physics, the specific case of vectors in 3-D is so prevalent that it warrants special attention, and also serves as an introduction to more general formulations A 3-D vector is defined as an entity that has both magnitude and direction, and can be characterized, provided a basis is specified, by an ordered triple of numbers The vector x, then, is represented as x = (x1 , x2 , x3 ) Consider the following definitions for operations on the vectors x and y given by x = (x1 , x2 , x3 ) and y = (y1 , y2 , y3 ): Vector equality: x = y if xi = yi ∀ i = 1, 2, Vector addition: x + y = z if zi = xi + yi ∀ i = 1, 2, 3 Scalar multiplication: ax = (ax1 , ax2 , ax3 ) Null vector: There exists a unique null vector = (0, 0, 0) CHAPTER VECTOR CALCULUS 1.2 FUNDAMENTAL OPERATIONS INVOLVING VECTORS Furthermore, assume that the following properties hold for the above defined operations: Vector addition is commutative and associative: x+y =y+x (x + y) + z = x + (y + z) Scalar multiplication is associative and distributive: (ab)x = a(bx) (a + b)(x + y) = ax + bx + ay + by The collection of all 3-D vectors that satisfy the above properties are said to form a 3-D vector space 1.2 Fundamental operations involving vectors The following fundamental vector operations are defined Scalar Product: a · b = ax bx + ay by + az bz = bi = |a||b|cos(θ) i = b·a (1.1) √ where |a| = a · a, and θ is the angle between the vectors a and b Cross Product: ˆ (ay bz − az by ) + y ˆ (az bx − ax bz ) + z ˆ(ax by − ay bx ) a×b=x (1.2) or more compactly c = a×b where ci = (1.3) (1.4) aj bk − ak bj (1.5) where i, j, k are x, y, z and the cyclic permutations z, x, y and y, z, x, respectively The cross product can be expressed as a determinant: The norm of the cross product is |a × b| = |a||b|sin(θ) (1.6) where, again, θ is the angle between the vectors a and b The cross product of two vectors a and b results in a vector that is perpendicular to both a and b, with magnitude equal to the area of the parallelogram defined by a and b The Triple Scalar Product: a·b×c=c·a×b=b·c×a (1.7) and can also be expressed as a determinant The triple scalar product is the volume of a parallelopiped defined by a, b, and c The Triple Vector Product: a × (b × c) = b(a · c) − c(a · b) (1.8) The above equation is sometimes referred to as the BAC − CAB rule Note: the parenthases need to be retained, i.e a × (b × c) = (a × b) × c in general Lattices/Projection of a vector a = (ax )ˆ x + (ay )ˆ y + (ay )ˆ y (1.9) 1.2 FUNDAMENTAL OPERATIONS INVOLVING VECTORS CHAPTER VECTOR CALCULUS a·x ˆ = ax (1.10) r = r1 a1 + r2 a2 + r3 a3 (1.11) · aj = δij (1.12) = aj × ak · (aj × ak ) (1.13) Gradient, ∇ ˆ ∇=x ∂ ∂ ∂ ˆ ˆ +y +z ∂x ∂y ∂z ∇f (|r|) = ˆr ∂f ∂r (1.14) (1.15) ˆ dx + y ˆ dy + z ˆdz dr = x (1.16) dϕ = (∇ϕ) · dr (1.17) ∇(uv) = (∇u)v + u(∇v) (1.18) Divergence, ∇· ∇·V = ∂Vy ∂Vx ∂Vz + + ∂x ∂y ∂z ∇·r=3 ∇ · (rf (r)) = 3f (r) + r (1.19) (1.20) df dr (1.21) if f (r) = rn−1 then ∇ · ˆrrn = (n + 2)rn−1 ∇ · (f v) = ∇f · v + f ∇ · v (1.22) Curl,∇× ˆ( x ∂Vy ∂Vy ∂Vz ∂Vx ∂Vz ∂Vx ˆ( ˆ( − )+y − )+z − ) ∂y ∂z ∂z ∂x ∂x ∂y (1.23) ∇ × (f v) = f ∇ × v + (∇f ) × v (1.24) ∇×r=0 (1.25) ∇ × (rf (r)) = (1.26) ∇(a · b) = (b · ∇)a + (a · ∇)b + b × (∇ × a) + a × (∇ × b) (1.27) CHAPTER VECTOR CALCULUS 1.2 FUNDAMENTAL OPERATIONS INVOLVING VECTORS ∂2 ∂2 ∂2 + + ∂2x ∂2y ∂2z ∇ · ∇ = ∇ × ∇ = ∇2 = (1.28) Vector Integration Divergence theorem (Gauss’s Theorem) ∇ · f (r)d3 r = V ∇ · f (r) · dσ = S ∇ · f (r) · nda (1.29) S let f (r) = u∇v then ∇ · (u∇v) = ∇u · ∇v + u∇2 v ∇u · ∇vd3 r + V u∇2 vd3 r = V (1.30) (u∇v) · nda (1.31) S The above gives the second form of Green’s theorem Let f (r) = u∇v − v∇u then ∇u · ∇vd3 r + V u∇2 vd3 r − V ∇u · ∇vd3 r − V v∇2 ud3 r = V (u∇v) · nda − S (v∇u) · nda (1.32) S Above gives the first form of Green’s theorem Generalized Green’s theorem ˆ − uLvd ˆ 3r = uLu V p(v∇u − u∇v)) · nda (1.33) S ˆ is a self-adjoint (Hermetian) “Sturm-Lioville” operator of the form: where L ˆ = ∇ · [p∇] + q L (1.34) Stokes Theorem (∇ × v) · nda = S V · dλ (1.35) dλ ◦ [] (1.36) C Generalized Stokes Theorem (dσ × ∇) ◦ [] = S C where ◦ = ,·,× Vector Formulas a · (b × c) = b · (c × a) = c · (a × b) (1.37) a × (b × c) = (a · c)b − (a · b)c (1.38) (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c) (1.39) ∇ × ∇ψ = (1.40) ∇ · (∇ × a) = (1.41) 1.2 FUNDAMENTAL OPERATIONS INVOLVING VECTORS CHAPTER VECTOR CALCULUS ∇ × (∇ × a) = ∇(∇ · a) − ∇2 a (1.42) ∇ · (ψa) = a · ∇ψ + ψ∇ · a (1.43) ∇ × (ψa) = ∇ψ × a + ψ∇ × a (1.44) ∇(a · b) = (a · ∇)b + (b · ∇)a + a × (∇ × b) + b × (∇ × a) (1.45) ∇ · (a × b) = b · (∇ × a) − a · (∇ × b) (1.46) ∇ × (a × b) = a(∇ · b) − b(∇ · a) + (b · ∇)a − (a · ∇)b (1.47) If x is the coordinate of a point with magnitude r = |x|, and n = x/r is a unit radial vector ∇·x=3 (1.48) ∇×x=0 (1.49) ∇ · n = 2/r (1.50) ∇×n=0 (1.51) (a · ∇)n = (1/r)[a − n(a · n)] (1.52) 10 13.8 COMPLEX FORM OF THE FOURIER SERIES CHAPTER 13 EXAPANSIONS 13.8 Complex Form of the Fourier Series Suppose that f (x) is either a real or complex function which is piecewise smooth on xo − L ≤ x ≤ xo + L The Fourier series for f (x), valid on that interval, can be written: f (x) = Ao + = Ao + = Ao + ∞ An cos n=1 ∞ An n=1 ∞ n=1 e nπx nπx + Bn sin L L inπx L + e− inπx L e + Bn inπx L − e− inπx L inπx inπx 1 (An − iBn )e L + (An + iBn )e− L 2 (13.28) Let xo +L 1 cn ≡ (An − iBn ) = 2L cn = cos xo −L xo +L 2L nπx nπx − c sin f (x)dx L L f (x)e− inπx L dx (13.29) f (x)dx = co (13.30) xo −L Then 1 Ao = 2L xo +L xo −L and 1 (An + iBn ) = 2L = 2L xo +L f (x) cos xo −L xo +L f (x)e inπx L nπx nπx + i sin dx L L dx xo −L =C−n (13.31) Hence ∞ f (x) = co + cn e inπx L + c−n e− inπx L (13.32) n=1 or ∞ f (x) = cn e inπx L xo − L ≤ x ≤ xo + L n=−∞ This is called the complex form of the Fourier series for f (x), whether or not f (x) itself is complex It converges to [f (x−) + f (x+)] for xo − L < x < xo − L At x = xo − L and x = xo + L, it converges to [f (xo 130 − L+) + f (xo + L−)] CHAPTER 13 EXAPANSIONS 13.9 UNIFORM CONVERGENCE OF FOURIER SERIES 13.9 Uniform Convergence of Fourier Series f (x) = Ao + ∞ [An cos nx + Bn sin nx] (13.33) n=1 (|A| + |B|)2 =|A|2 + |B|2 + 2|A| |B| (|A| − |B|)2 =|A|2 + |B|2 − 2|A| |B| (|A| + |B|)2 + (|A| − |B|)2 =2(|A|2 + |B|2 ≥ (|A| + |B|)2 hence |A| + |B| ≤ √ 2(|A|2 + |B|2 ) |An cos nx + Bn sin nx| ≤|An cos nx| + |Bn sin nx| ≤|An | + |Bn | ≤[|An |2 + |Bn |2 ] √ = 2[A2n + Bn2 ] taking f (x) to be real, so the An and Bn are real Hence the Fourier series for f (x) converges absolutely and 2 21 uniformly if the series N n=1 [An + Bn ] converges Let N [A2n + Bn2 ] SN ≡ n=1 N = n=1 N 1 2 [n (An + Bn2 )] n = n=1 n2 [n2 (A2n + Bn2 )] |Dn | |cn | N ≤ n=1 n2 N n2 (A2n + Bn2 ) n=1 |(c, D)| ≤ c D from the Schwarz inequality in a vector space in which vectors are N-Tuples of real numbers, such as c ≡ N (c1 , c2 cN ), d ≡ (d1 , d2 dN , and the scalar product is defined by (c, d) ≡ n=1 cn dn The Schwarz 1 2 inequality is |(c, d)| ≤ (c, c) (d, d) , or N N c2n cn dn ≤ n=1 n=1 N d2n (13.34) n=1 With the choice cn = n 1 dn =[n2 (A2n + Bn2 )] = n(A2n + Bn2 ) 131 (13.35) 13.10 DIFFERENTIATION OF FOURIER SERIES N CHAPTER 13 EXAPANSIONS N (A2n + Bn2 ) ≤ n=1 n=1 n2 N n2 (A2n + Bn2 ) (13.36) n=1 If, for −π ≤ x ≤ π, f (x) is continuous, f (x) is piecewise continuous, and f (−π) = f (π), then the series also converges, it follows ∞ ∞ 2 n=1 n (An + Bn ) converges, as shown in problem (9) Since the series n=1 n2 2 that ∞ n=1 (An + Bn ) converges, so the Fourier series for f (x) converges uniformly ∞ n=1 ∞ n=1 and absolutely π2 = n2 =ζ(p) np (Reimann Zeta) Hence we have proven: Theorem If f (x) is continuous and if it is only piecewise continuous and f (x) is piecewise continuous for −π ≤ x ≤ π, and f (−π) = f (π), then the Fourier series for f (x) converges uniformly and absolutely on −π ≤ x ≤ π (or any interval that is continuous) It can be shown, by a more complicated proof, that the Fourier series for a piecewise smooth function converges uniformly and absolutely in any closed subinterval in which the function is continuous Piecewise smooth ≡ f (x), f (x) are piecewise continuous 13.10 Differentiation of Fourier Series If f (x) is periodic and continuous, and if f (x) and f (x) are piecewise continuous, then the term-by-term derivative of the Fourier series for f (x) equals f (x) except at points where f (x) is discontinuous Under these conditions both f (x) and f (x) have convergent Fourier series, and the term-by-term derivation of the series for f (x) equals the series for f (x) Details are worked out in problem (12.) 13.11 Integration of Fourier Series nπx nπx If f (x) is piecewise continuous for −L ≤ x ≤ L, then, whether or not the Fourier series A2o + ∞ n=1 An cos L + Bn sin L x corresponding to f (x) converges, the term-by-term integral of the series from −L to x ≤ L equals −L f (x)dx Proof: x F (x) ≡ f (x)dx − Ao x −L is a continuous function, with derivative F (x) = f (x) − 21 Ao which is piecewise continuous on −L ≤ x ≤ L for f (x) piecewise continuous on −L ≤ x ≤ L Show F (x) periodic 132 CHAPTER 13 EXAPANSIONS 13.11 INTEGRATION OF FOURIER SERIES Also F (−L) = 12 Ao L, and L f (x)dx − Ao L −L =LAo − Ao L = Ao L = F (−L) F (L) = Hence F (x) has a Fourier series ao F (x) = + ∞ an cos n=1 nπx nπx + bn sin L L (13.37) which converges to f (x) for all points on −L ≤ x ≤ L At x = L, F (L) = Ao L ∞ ao = + an cos nπ +bn sin nπ n=1 = ao + (−1)n ∞ an (−1)n n=1 Hence ao = Ao L − 2 ∞ an (−1)n (13.38) n=1 Also, for n > an = = L L =− =− L F (x) cos −L nπx dx L F (x) sin nπx L nπ L nπ nπ L −L L F (x) sin −L L −L L nπ L L =− Bn nπ L F (x) sin nπx L −L nπ L − nπx dx L nπx f (x) − Ao sin dx L L =− f (x) sin −L nπx dx L for n > 133 dx 13.11 INTEGRATION OF FOURIER SERIES CHAPTER 13 EXAPANSIONS And bn = = = = L L F (x) sin −L L F (x) nπx dx L L − cos nπx L nπ L L nπ L − cos nπx L nπ L nπx f (x) − Ao cos dx L −L L −L sin nπx nπx L f (x) cos dx − Ao nπ L −L L L L = nπ L L = An nπ dx nπx dx L F (x) cos −L L F (x) −L −L L L nπ L L − for n > Hence the Fourier series F (x) = ao + ∞ an cos n=1 nπx nπx + bn sin L L (13.39) becomes F (x) = Ao L − = Ao L + = Ao L + But F (x) ≡ x −L f (x)dx ∞ ∞ n an (−1) n=1 ∞ + an cos n=1 an cos n=1 ∞ − n=1 nπx nπx + bn sin L L nπx nπx − (−1)n + bn sin L L L nπx L nπx Bn cos − (−1)n + An sin nπ L nπ L − 21 Ao x, so x f (x)dx = Ao (x + L) + −L x = −L Ao dx + ∞ An n=1 ∞ L nπx L nπx sin − Bn cos − (−1)n nπ L nπ L x An n=1 cos −L nπx dx + Bn L x sin −L nπx dx L =Term-by-term integral of the Fourier series for f (x) f (x) = fourier series for f f (x)piecewise continuous f (x)continuous ∂ ∂f (x) = f s ∂x ∂x f (x)periodic and or piecewise continuous f (x)p.c f (x)p.c f (x) = four ser f (x)p.c 134 (13.40) CHAPTER 13 EXAPANSIONS 13.12 FOURIER INTEGRAL REPRESENTATION 13.12 Fourier Integral Representation If f (x) is piecewise smooth for −L ≤ x ≤ L, it has a complex Fourier series ∞ πx cn ein L F (x) = cn ≡ n=−∞ 2L L πx f (x)e−in L dx −L which converges to 12 [f (x+) + f (x−)] for −L < x < L We wish to consider what happens as L → ∞ We cannot naively put l = ∞ for cn because that gives ∞ ∞ cn = 2π n=−∞ cn Putting cn in the series for f (x): −∞ f (x)dx for all n, and the Fourier series becomes just ∞ n=−∞ ∞ = Let ∆k ≡ π L L 2L f (x) = f (x ) ei nπx L −L L 2L n=−∞ nπ dx f (x )ei L (x−x ) (13.41) dx f (x )ei nδk(x−x ) (13.42) −L Then, ∞ L ∆k 2π n=−∞ f (x) = But −L ∞ ∞ lim ∆k→0 Hence, as L → ∞, ∆k ≡ π L F (k)dk F (n∆k)∆k = (13.43) −∞ n→−∞ → Fourier integral expression for f (x): 2π f (x) = ∞ ∞ dx f (x )eik(x−x ) dk −∞ (13.44) −∞ This proof is heuristic (suggestive, but not rigorous) The result can be written ∞ f (x) = √ 2π where F (k) ≡ √ 2π ∞ dkF (k)eikx (13.45) −∞ dx f (x )e−ikx Fourier transform of f (x) −∞ Before providing rigorously the Fourier integral theorem, we need to consider “improper integrals” of the form: ∞ I(k) = f (x, k)dx a R = lim R→∞ a f (x, k)dx IR (k) = lim IR (k) R→∞ or 135 (13.46) 13.13 M-TEST FOR UNIFORM CONVERGENCE CHAPTER 13 EXAPANSIONS ∞ I(k) = f (x, k)dx −∞ R = lim R→∞ −R F (x, k)dx = lim IR (k) (13.47) R→∞ I(k) is said to converge uniformly in k, for A ≤ k ≤ B, if given > o, one can find Q, independent of k for a ≤ k ≤ b, such that |I(k) − IR (k)| < for R > Q 13.13 M-Test for Uniform Convergence ∞ ∞ If |f (x, k)| ≤ M (x) for A ≤ k ≤ B and if a M (x) exists, then a f (x, k)dx converges uniformly A uniformly convergent improper integral I(k) can be integrated under the integral sign: ∞ B dk A ∞ B f (x, k)dx = f (x, k)dk dx −∞ −∞ (13.48) A These properties are analogous to those of a uniformly convergent infinite series: ∞ S(k) = Fn (k) n=0 N = lim Fn (k) (13.49) F (x)eikx dx (13.50) N →∞ An important simpler example is n=0 ∞ I(k) = −∞ ∞ If −∞ |f (x)|dx exists, then I(k) converges uniformly in k, since |f (x)eikx | = |f (x)|, so one can take M (x) = |f (x)| 13.14 Fourier Integral Theorem If f (x) is piecewise smooth for all finite intervals, if ∞ −∞ |f (x)|dx ∞ F (k) ≡ √ 2π then exists, and if f (x)e−ikx dx (13.51) −∞ 1 f (x) ≡ [f (x+) + f (x−)] = √ 2π F{ } =√ 2π F −1 { } = √ 2π ∞ −∞ ∞ −∞ Proof: Reference: Dettman, Chapter 8, or Churchill 136 ∞ F (k)eikx dk −∞ dxe−ikx { } dke−ikx { } (13.52) CHAPTER 13 EXAPANSIONS 13.14 FOURIER INTEGRAL THEOREM Lemma Riemann’s Lemma If f (x) is piecewise continuous for a ≤ x ≤ b, a and b finite, then limR→∞ b a f (x) sin Rx d Proof: Since the integral can be split up into integrals of a continuous fct, Lemma is true if it si true for a continuous f (x), which we now assume If we change the variable from x to t such that Rx = Rt + π, so π x=t+ R , π b− R b f (x) sin Rxdx = f t+ π a− R a − sin Rt π b− R =− π sin(Rt + π) dt R f t+ π a− R π sin Rt dt R (13.53) Hence, adding these expressions: b π b− R b f (x) sin Rx dx − f (x) sin Rx dx = a π a− R a a =− − f x+ f x+ π a− R π b− R π sin Rx dx R f x+ a π sin Rx dx R π − f (x) sin Rx dx R b + f (x) sin Rx dx π b− R b a f (x) sin Rx dx ≤ f x+ π a− R π b− R a + π sin Rx dx R f x+ a π − f (x) R b + f (x) sin Rx dx π b− R π π π ≤ M + b− −a f x+ − f (x) R R R π for a ≤ x ≤ b R where |f (x)| ≤ M for a ≤ x ≤ b M is finite, and f x + π R + max π M R − f (x) → as R → ∞ max because f (x) is continuous Hence b lim R→∞ f (x) sin Rxdx = a b ⇒ lim R→∞ a f (x) sin Rxdx = Lemma If f (x) is piecewise smooth in any finite interval and T lim R→∞ −T f (x + t) ∞ −∞ |f (x)|dx sin Rt π dt = [f (x+) + f (x−)] t 137 < ∞, then 0 can find T large enough so + f (t)dt < Hence ∞ sin Rt π lim f (x + t) dt − [f (x+) + f (x−)] R→∞ −∞ t T < + lim R→∞ f (x + t) −T 0, sin Rt π dt − [f (x+) + f (x−)] t from proof for finite T , Lemma =0, arbitrary, since Q.E.D ∞ ⇒ lim R→∞ −∞ f (x + t) π sin Rt dt = [f (x+) + f x−] t Proof of the Fourier Integral Theorem: √ 2π ∞ F (k)eikx dk = √ 2π −∞ =√ 2π ∞ dk √ 2π −∞ ∞ ∞ f (x )e−ikx dx −∞ ∞ dx f (x )eik(x−x ) dk −∞ = lim √ R→∞ 2π 138 −∞ R ∞ dx f (x )eik(x−x ) dk −R −∞ eikx (13.54) CHAPTER 13 EXAPANSIONS 13.15 EXAMPLES OF THE FOURIER INTEGRAL THEOREM ∞ But |f (x )eik(x−x ) | = |f (x )| and −∞ dx |f (x )| < ∞, so the integral over x converges uniformly in k, so the order of the integration can be exchanged √ 2π ∞ R→∞ 2π ∞ R F (x)eikx dk = lim −∞ = lim R→∞ 2π dk eik(x−x ) dx f (x ) −∞ ∞ −R dx f (x ) −∞ sin k(x − x ) (x − x ) ∞ sin R(x − x ) = lim dx f (x ) R→∞ π −∞ x−x ∞ sin Rt = lim dt f (x + t) R→∞ π −∞ t = [f (x+) + f (x−)] from Lemma k=R k=−R Q.E.D We note for future reference that the Fourier transform F (k) of a piecewise smooth function f (x) statisfying < ∞ is bounded: ∞ −∞ |f (x)|dx F (k) ≡ √ 2π ∞ f (x)e−ikx dx −∞ so ∞ |F (k)| ≤ √ |f (x)e−ikx |dx 2π −∞ ∞ =√ |f (x)|dx 2π −∞ =B < ∞ since ∞ −∞ |f (x)|dx σ sin(k − ko )σ π (k − ko )σ π =π σ Parseval’s Theorem for Fourier Transforms ∞ ∞ Suppose f (x) and g(x) are piecewise smooth for all finite intervals, and −∞ |f (x)|dx and −∞ |g(x)|dx exist, so f (x) and g(x) have Fourier integral representations with Fourier transforms F (k) and G(k) which are bounded, so in particular |G(k)| < B for all k The Parseval theorem is that: ∞ g ∗ (x)f (x)dx = −∞ ∞ −∞ 141 G∗ (k)F (k)dk (13.59) 13.17 CONVOLUTION THEOREM FOR FOURIER TRANSFORMS CHAPTER 13 EXAPANSIONS Proof: ∞ G∗ (k)F (k)dk = −∞ ∞ √ 2π dkG∗ (k) −∞ ∞ ∞ =√ 2π dk −∞ ∞ f (x)e−ikx dx −∞ f (x)G∗ (k)e−ikx dx −∞ But the integral over x converges uniformly in k, since |f (x)G∗ (k)e−ikx | ≤ |f (x)|B, and ∞ ≡ M (x) ∞ |f (x)|dx < ∞ M (x)dx = B −∞ ∞ Hence the order of integration can be changed: ∞ ∞ G∗ (k)F (k)dk = √ 2π −∞ ∞ dxF (x) −∞ ∞ = √ 2π dxF (x) −∞ ∞ = dxf (x)g(x)∗ , G∗ (k)e−ikx dk −∞ ∞ (13.60) ∗ ikx G(k)e dk −∞ Q.E.D −∞ For the special case g(x) = f (x), ∞ ∞ |f (x)|2 dx = |F (k)|2 dk −∞ 13.17 (13.61) −∞ Convolution Theorem for Fourier Transforms If H(k) = G(k)F (k), where G(k) = √ 2π ∞ g(x)e−ikx dx and −∞ F (k) = √ 2π ∞ f (x)e−ikx dx, −∞ then h(x) = √ 2π =√ 2π =√ 2π =√ 2π ∞ H(k)eikx dk −∞ ∞ G(k)F (k)eikx dk −∞ ∞ dk −∞ ∞ g(x )e−ikx dx F (k)eikx −∞ ∞ dx g(x )eik(x−x ) (k) dk −∞ ∞ √ 2π −∞ The integral over x converges uniformly in k, since ∞ dx |g(x )| −inf ty 142 (13.62) CHAPTER 13 EXAPANSIONS 13.18 FOURIER SINE AND COSINE TRANSFORMS AND REPRESENTATIONS exists, and |F (k)| ≤ B, so the order of integration can be exchanged ∞ ∞ dx dkF (k)eik(x−x ) g(x ) 2π −∞ −∞ ∞ ∞ 1 =√ dx g(x ) √ F (k)eik(x−x ) dk 2π −∞ 2π −∞ h(x) = h(x) = √ 2π (13.63) ∞ g(x )f (x − x )dx ; Called the Fourier Convolution of f (x) and g(x) −∞ If we change the variable of integration to x = x − x , dx = −dx h(x) = √ 2π =√ 2π =√ 2π −∞ g(x − x )f (x )(−dx ) ∞ ∞ f (x )g(x − x )dx −∞ ∞ f (x )g(x − x )dx (13.64) −∞ Hence the convolution is symmetric in the two functions f and g The significance of the convolution h(x) is that it is the function whose Fourier transform is the product of the transforms of f and g ¯ F {h(x)} = h(k) = g¯(k)f¯(k) 13.18 Fourier Sine and Cosine Transforms and Representations F (k) ≡ √ 2π ∞ f (x)e−ikx dx = √ 2π −∞ Hence, if f (−x) = f (x), an even function, 143 ∞ f (x)[cos kx − sin kx]dx −∞ Bibliography (1) A RFKEN , G B., AND W EBER , H J Mathematical methods for physicists, ed Academic Press, San Diego, 2001 (2) BAMBERG , P., AND S TERNBERG , S A course in mathematics for students of physics: Cambridge University Press, Cambridge, 1991 (3) BAMBERG , P., AND S TERNBERG , S A course in mathematics for students of physics: Cambridge University Press, Cambridge, 1991 (4) C HOQUET-B RUHAT, Y., D E W ITT-M ORETTE , C., AND D ILLARD -B LEICK , M Analysis, manifolds and physics Part I: basics, revised ed North-Holland, Amsterdam, 1991 (5) C HOQUET-B RUHAT, Y., AND D E W ITT-M ORETTE , C Analysis, manifolds and physics Part II: 92 applications North-Holland, Amsterdam, 1989 (6) A LLEN , M., AND T ILDESLEY , D Computer Simulation of Liquids Oxford University Press, 1987 144 ... =1T = A1 = 1A = A Commutator: a linear operation [A, B] ≡ AB − BA [A, B] = if A and B are diagonal matrices Diagonal Matrices: Aij = aii δij Jacobi Identity: [A, [B, C]] = [B, [A, C]] − [C, [A, B]]... orthogonal matrix is unitary or a real unitary matrix is orthogonal (if a matrix is real, unitary=orthogonal) 2.15 Comments about Hermitian Matrices and Unitary Tranformations Unitary tranformations... then [A, A+ ] = and conversely 21 normal Chapter Calculus of Variations 3.1 Functions and Functionals Here we consider functions and functionals of a single argument (a variable and function, respectively)