Abstract: Course material for mathemathical methods of theoretical physics intended for an undergraduate audience.

Table of Contents

• 1 Unreasonable effectiveness of mathematics in the natural sciences 23
• 2 Methodology and proof methods 27
• 3 Numbers and sets of numbers 31
• Part II: Linear vector spaces 33
• 4 Finite-dimensional vector spaces 35
• 4.1 Basic definitions 35
• —4.1.1 Fields of real and complex numbers, 35.
• —4.1.2 Vectors and vector space, 36.
• 4.2 Linear independence 37
• 4.3 Subspace 37
• —4.3.1 Scalar or inner product [§61], 37.
• —4.3.2 Hilbert space, 39.
• 4.4 Basis 39
• 4.5 Dimension [§8] 39
• 4.6 Coordinates [§46] 40
• 4.7 Finding orthogonal bases from nonorthogonal ones 42
• 4.8 Mutually unbiased bases 43
• 4.9 Direct sum 44
• 4.10 Dual space 44
• —4.10.1 Dual basis, 45.
• —4.10.2 Dual coordinates, 47.
• —4.10.3 Representation of a functional by inner product, 48.
• 4.11 Tensor product 48
• —4.11.1 Definition, 48.
• —4.11.2 Representation, 49.
• 4.12 Linear transformation 49
• —4.12.1 Definition, 49.
• —4.12.2 Operations, 49.
• —4.12.3 Linear transformations as matrices, 50.
• 4.13 Projector or Projection 51
• —4.13.1 Definition, 51.
• —4.13.2 Construction of projectors from unit vectors, 52.
• 4.14 Change of basis 53
• 4.15 Rank 54
• 4.16 Determinant 54
• —4.16.1 Definition, 54.
• —4.16.2 Properties, 54.
• 4.17 Trace 55
• —4.17.1 Definition, 55.
• —4.17.2 Properties, 55.
• 4.18 Adjoint 55
• —4.18.1 Definition, 55.
• —4.18.2 Properties, 56.
• —4.18.3 Matrix notation, 56.
• 4.19 Self-adjoint transformation 56
• 4.20 Positive transformation 57
• 4.21 Unitary transformations and isometries 57
• —4.21.1 Definition, 57. 
• —4.21.2 Characterization of change of orthonormal basis,57.
• —4.21.3 Characterization in terms of orthonormal basis, 58.
• 4.22 Orthogonal projectors 58
• 4.23 Proper value or eigenvalue 58
• —4.23.1 Definition, 58.
• —4.23.2 Determination, 58.
• 4.24 Normal transformation 62
• 4.25 Spectrum 62
• —4.25.1 Spectral theorem, 62.
• —4.25.2 Composition of the spectral form, 62.
• 4.26 Functions of normal transformations 63
• 4.27 Decomposition of operators 64
• —4.27.1 Standard decomposition, 64.
• —4.27.2 Polar representation, 64.
• —4.27.3 Decomposition of isometries, 64.
• —4.27.4 Singular value decomposition, 64.
• —4.27.5 Schmidt decomposition of the tensor product of two vectors, 65.
• 4.28 Commutativity 66
• 4.29 Measures on closed subspaces 67
• —4.29.1 Gleason’s theorem, 68.
• —4.29.2 Kochen-Specker theorem, 68.
• 4.30 Hilbert space quantummechanics and quantumlogic 69
• —4.30.1 Quantummechanics, 69.
• —4.30.2 Quantum logic, 72.
• —4.30.3 Diagrammatical representation, blocks, complementarity, 74.
• —4.30.4 Realizations of two-dimensional beam splitters, 75.
• —4.30.5 Two particle correlations, 78.
• 5 Tensors 85
• 5.1 Notation 85
• 5.2 Multilinear form 86
• 5.3 Covariant tensors 86
• —5.3.1 Basis transformations, 87.
• —5.3.2 Transformation of Tensor components, 88.
• 5.4 Contravariant tensors 88
• —5.4.1 Definition of contravariant basis, 88.
• —5.4.2 Connection between the transformation of covariant and contravariant entities, 89.
• 5.5 Orthonormal bases 90
• 5.6 Invariant tensors and physical motivation 90
• 5.7 Metric tensor 90
• —5.7.1 Definition metric, 90.
• —5.7.2 Construction of a metric from a scalar product by metric tensor, 90
• —5.7.3 What can themetric tensor do for us?, 91.
• —5.7.4 Transformation of the metric tensor, 92.
• —5.7.5 Examples, 92.
• —5.7.6 Decomposition of tensors, 95.
• —5.7.7 Form invariance of tensors, 95.
• 5.8 The Kronecker symbol ± 100
• 5.9 The Levi-Civita symbol ” 100
• 5.10 The nabla, Laplace, and D’Alembert operators 101
• 5.11 Some tricks 101
• 5.12 Some common misconceptions 102
• —5.12.1 Confusion between component representation and “the real thing”, 102.
• —5.12.2 A matrix is a tensor, 102.
• Part III: Functional analysis 109
• 6 Brief review of complex analysis 111
• 6.1 Differentiable, holomorphic (analytic) function 112
• 6.2 Cauchy-Riemann equations 112
• 6.3 Definition analytical function 112
• 6.4 Cauchy’s integral theorem 113
• 6.5 Cauchy’s integral formula 114
• 6.6 Laurent series 115
• 6.7 Residue theorem 116
• 6.8 Multi-valued relationships, branch points and and branch cuts 119
• 6.9 Riemann surface 119
• 7 Brief review of Fourier transforms 121
• 8 Distributions as generalized functions 125
• 8.1 Heaviside step function 129
• 8.2 The sign function 130
• 8.3 Useful formulæ involving ± 131
• 8.4 Fourier transforms of ± and H 133
• 9 Green’s function 141
• 9.1 Elegant way to solve linear differential equations 141
• 9.2 Finding Green’s functions by spectral decompositions 143
• 9.3 Finding Green’s functions by Fourier analysis 145
• Part IV: Differential equations 151
• 10 Sturm-Liouville theory 153
• 10.1 Sturm-Liouville form 153
• 10.2 Sturm-Liouville eigenvalue problem 154
• 10.3 Adjoint and self-adjoint operators 155
• 10.4 Sturm-Liouville transformation into Liouville normal form 156
• 10.5 Varieties of Sturm-Liouville differential equations 158
• 11 Separation of variables 161
• 12 Special functions of mathematical physics 165
• 12.1 Gamma function 165
• 12.2 Beta function 167
• 12.3 Fuchsian differential equations 167
• —12.3.1 Regular, regular singular, and irregular singular point, 168.
• —12.3.2 Power series solution, 169.
• 12.4 Hypergeometric function 180
• —12.4.1 Definition, 180.
• —12.4.2 Properties, 182.
• —12.4.3 Plasticity, 183.
• —12.4.4 Four forms, 186.
• 12.5 Orthogonal polynomials 186
• 12.6 Legendre polynomials 187
• —12.6.1 Rodrigues formula, 187.
• —12.6.2 Generating function, 188.
• —12.6.3 The three term and other recursion formulae, 188.
• —12.6.4 Expansion in Legendre polynomials, 190.
• 12.7 Associated Legendre polynomial 191
• 12.8 Spherical harmonics 192
• 12.9 Solution of the Schrödinger equation for a hydrogen atom 192
• —12.9.1 Separation of variables Ansatz, 193.
• —12.9.2 Separation of the radial part from the angular one, 193.
• —12.9.3 Separation of the polar angle µ from the azimuthal angle ‘, 194.
• —12.9.4 Solution of the equation for the azimuthal angle factor ©(‘), 194.
• —12.9.5 Solution of the equation for the polar angle factor £(µ), 195.
• —12.9.6 Solution of the equation for radial factor R(r ), 197.
• —12.9.7 Composition of the general solution of the Schrödinger Equation, 199.
• 13 Divergent series 201
• 13.1 Convergence and divergence 201
• 13.2 Euler differential equation 202
• Bibliography 205