Advanced Theoretical Physics

Description

[The book is self-published by the author. Check out his Youtube and website, where you can purchase this book for any price you choose: https://scienceasylum.com/ Of course you can still download it here.] Are you an undergraduate physics student planning on attending graduate school? Are you a graduate physics student already, but you feel like you're missing something? Either way, this book could be a huge help! There are a lot of little bits of information that don’t get taught in undergraduate physics for one reason or another, but your graduate teachers still expect you to know it. That’s what this book is: all the super important bits you should know before entering graduate physics. This book is not intended for anyone without at least background in basic calculus and introductory physics, but if you like a challenge, then go for it! Inside, you’ll find mathematical topics like vector calculus and tensor analysis as well as physics topics like electrodynamics, relativity, and quantum mechanics. Historical context is also given throughout the text to give you a deeper insight into each topic and to show how long it took the scientific community to develop the ideas, which can make you feel a little better about maybe not understanding it right away. You can find a table of contents and some. You can also: buy the book watch on youtube listen to the podcast Preface ix 1 Coordinate Systems 1 1.1 Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Polar and Cylindrical . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Spherical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Bipolar and Elliptic . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Vector Algebra 11 2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Vector Calculus 19 3.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Del Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Non-Cartesian Del Operators . . . . . . . . . . . . . . . . . . 24 3.4 Arbitrary Del Operator . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Vector Calculus Theorems . . . . . . . . . . . . . . . . . . . . 36 The Divergence Theorem . . . . . . . . . . . . . . . . . . . 37 The Curl Theorem . . . . . . . . . . . . . . . . . . . . . . 39 4 Lagrangian Mechanics 45 4.1 A Little History... . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Derivation of Lagrange’s Equation . . . . . . . . . . . . . . . . 46 4.3 Generalizing for Multiple Bodies . . . . . . . . . . . . . . . . . 51 4.4 Applications of Lagrange’s Equation . . . . . . . . . . . . . . 52 4.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 Applications of Lagrange Multipliers . . . . . . . . . . . . . . 68 4.7 Non-Conservative Forces . . . . . . . . . . . . . . . . . . . . . 75 5 Electrodynamics 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Experimental Laws . . . . . . . . . . . . . . . . . . . . . . . . 77 Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . . . 78 Biot-Savart Law. . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Theoretical Laws . . . . . . . . . . . . . . . . . . . . . . . . . 97 Amp´ere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 97 Faraday’s Law. . . . . . . . . . . . . . . . . . . . . . . . . 105 Gauss’s Law(s) . . . . . . . . . . . . . . . . . . . . . . . . 108 Amp´ere’s Law Revisited . . . . . . . . . . . . . . . . . . . 111 5.4 Unification of Electricity and Magnetism . . . . . . . . . . . . 114 5.5 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . 118 5.6 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . 123 Maxwell’s Equations with Potentials. . . . . . . . . . . . . 127 5.7 Blurring Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6 Tensor Analysis 131 6.1 What is a Tensor? . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.3 Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4 Describing a Space . . . . . . . . . . . . . . . . . . . . . . . . 141 Line Element . . . . . . . . . . . . . . . . . . . . . . . . . 141 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 141 Raising and Lowering Indices . . . . . . . . . . . . . . . . 142 Coordinate Basis vs. Orthonormal Basis. . . . . . . . . . . 143 6.5 Really... What’s a Tensor?! . . . . . . . . . . . . . . . . . . . . 144 6.6 Coordinate Transformations . . . . . . . . . . . . . . . . . . . 149 6.7 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7 Special Relativity 167 7.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.2 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Line Element . . . . . . . . . . . . . . . . . . . . . . . . . 170 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 172 Coordinate Rotations . . . . . . . . . . . . . . . . . . . . . 173