Mathematical Methods of Theoretical Physics

cab1729:

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
  • 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

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First Observation Of A Macroscopic Quantum Jump

(via spytap)

Physicists have watched an artificial atom jump from one state to another using a monitoring technique that could have important implications for quantum computing.

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