Theoretical Physics ReferenceRelease 0.5

Ondrej Certk

February 19, 2018

CONTENTS

1 Introduction 11.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Contributors 3

3 Mathematics 53.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Fourier Transform of a Periodic Function (e.g. in a Crystal) . . . . . . . . . . . . . . . . . . . . . . 343.6 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.7 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.9 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.12 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.13 Argument function, atan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.14 Multiple Argument Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.15 Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.16 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.17 Variations and Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.18 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.19 Homogeneous Functions (Eulers Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.20 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.21 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.22 Double Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.23 Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.24 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.25 Incomplete Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.26 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.27 Double Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.28 Fermi-Dirac Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.29 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.30 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.31 Gaunt Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.32 Wigner 3j Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.33 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.34 Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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3.35 Feynman Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.36 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.37 Wigner D Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.38 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.39 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.40 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.41 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4 Classical Mechanics, Special and General Relativity 1854.1 Gravitation and Electromagnetism as a Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.3 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5 Classical Electromagnetism 2255.1 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255.2 Semiconductor Device Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6 Thermodynamics and Statistical Physics 2536.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.2 Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

7 Fluid Dynamics 2637.1 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2637.2 MHD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2757.3 Compressible Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

8 Quantum Field Theory and Quantum Mechanics 2978.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.2 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.3 Quantum Electrodynamics (QED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3238.4 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3318.5 Systematic Perturbation Theory in QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3548.8 Radial Schrdinger and Dirac Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.9 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3768.10 Hartree-Fock (HF) Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.11 Projector Augmented-Wave Method (PAW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Bibliography 445

Index 447

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CHAPTER

ONE

INTRODUCTION

1.1 Preface

I have a very bad memory. I am able to memorize quite a lot of things short term, but I am not able to remember mostformulas from quantum mechanics over the long term (e.g. like over the summer). I dont remember formulas forperturbation theory (neither time dependent or time independent), I dont remember Feynman rules in quantum fieldtheory, I dont even remember the Dirac equation exactly (where the should be, if there is or 2, ...). The thingabout quantum field theory is not that some particular steps are difficult, but that there are so many of them and onehas to master all of them at once, in order to really get it.

I never got QFT, because once I mastered one part sufficiently, I forgot some other part and it took so much time tomaster that other part that I forgot the first part again. However, I was determined that I would get it. In order todo so, I realized I need to keep notes of things I understood, written in my own way. Then, when I relearn someparts that I forgot, it just takes me a few minutes to go over my reference notes to get into it quickly. My own styleof understanding is that the notes should be complete (no need to consult external books), yet very short and gettingdirectly to the point, and also with every single calculation carried out explicitly.

See also the preface to the QFT part.

If you want to study physics, learn math the physics way (as opposed to the usual mathematics way of a definition,theorem, proof, ...). When I was beginning my undergrad physics studies (and even on a high school), I also hadthis common misconception, that I need to study math and understand every proof and then Ill be somehow preparedfor physics. I was very wrong. I used to study calculus by myself and then trying to learn the proofs, and Lebesgueintegral and I was learning that from the mathematics books. At the university, I always did all my math exams first (asfar as I remember, I always got A from those), hoping that would be a good start for the physics exams, but I alwaysfound out that it was mostly useless.

Now I know that the only way to study physics is to go and do physics directly and learn the math on the way as needed.The math section of this book r