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Notes on Undergraduate Physics Lecture Notes on Undergraduate Physics Kevin Zhou [email protected] These notes review the undergraduate physics curriculum, with an emphasis on quantum mechanics. They cover, essentially, the material that every working physicist should know. The notes are not self-contained, but rather assume everything in the high school Physics Olympiad syllabus as prerequisite knowledge. The primary sources were: • David Tong's Classical Dynamics lecture notes. A friendly set of notes that covers Lagrangian and Hamiltonian mechanics with neat applications, such as the gauge theory of a falling cat. • Arnold, Mathematical Methods of Classical Mechanics. The classic advanced mechanics book. The first half of the book covers Lagrangian mechanics compactly, with nice and tricky problems, while the second half covers Hamiltonian mechanics geometrically. • Griffiths, Introduction to Electrodynamics. This is the definitive introduction to undergraduate electromagnetism: it's clear, comprehensive, and comes with great problems. There are also many neat references to the literature, especially in the 4th edition. As you'll see from following them, many subtle issues in classical electromagnetism were worked out by Griffiths himself! The only flaw of the book is that it largely sticks with traditional vector notation, even when index notation is clearer and faster. For an introduction to vector calculus that uses index notation, which is an essential tool in physics, see David Tong's Vector Calculus lecture notes. • David Tong's Electrodynamics lecture notes. Briefly covers electromagnetism at the same level as Griffiths, but freely uses index notation to simplify some tricky calculations, and perform some essential derivations that Griffiths avoids. • David Tong's Statistical Mechanics lecture notes. Has an especially good discussion of phase transitions, which leads in well to a further course on statistical field theory. • Blundell and Blundell, Concepts in Thermal Physics. A good first statistical mechanics book filled with applications, briefly touching on information theory, non-equilibrium thermodynam- ics, the Earth's atmosphere, and much more. • Sethna, Entropy, Order Parameters, and Complexity. An entertaining second statistical me- chanics book that touches on many foundational issues and modern topics. The exposition itself is very brief, but supplemented with extensive and useful exercises. 2 • Lautrup, Physics of Continuous Matter. A introduction to fluid dynamics and solids, with emphasis on real-world examples drawn from fields ranging from astrophysics to aerodynamics, and clear conceptual discussion. • David Tong's Applications of Quantum Mechanics lecture notes. A conversational set of notes, with a focus on solid state physics. Also contains a nice section on quantum foundations. • Griffiths, Introduction to Quantum Mechanics. This is the easiest book to start, but in college, I disliked it. The text is very chatty, full of distracting parentheticals and footnotes, but most of them are about the trivial details of calculations. The main purpose of the book seems to be to ensure that students are absolutely comfortable with the mechanics of solving the time-independent Schrodinger equation, but that means there's little coverage of the conceptual, historical, experimental, or mathematical background. But in retrospect, I think the book is very valuable because of its problems, which guide readers through important results, including some from the research literature and real-world applications. The 3rd edition also fixes some of the book's glaring omissions, such as by adding a chapter on symmetries. • Robert Littlejon's Physics 221 notes. An exceptionally clear set of graduate-level quantum mechanics notes, with a focus on atomic physics: you read it and immediately understand the world better than before. Complex material is developed elegantly, often in a cleaner and more rigorous way than in standard textbooks. These notes were developed as a replacement for Sakurai's Modern Quantum Mechanics, which becomes rather unclear in the latter half, and thus covers roughly the same topics. Much of my notes are just an imperfect summary of Littlejon's notes. The most recent version is here; please report any errors found to [email protected]. 3 Contents Contents 1 Classical Mechanics5 1.1 Lagrangian Formalism...................................5 1.2 Rigid Body Motion.....................................8 1.3 Hamiltonian Formalism.................................. 12 1.4 Poisson Brackets...................................... 14 1.5 Action-Angle Variables................................... 18 1.6 The Hamilton{Jacobi Equation.............................. 21 2 Electromagnetism 25 2.1 Electrostatics........................................ 25 2.2 Magnetostatics....................................... 27 2.3 Electrodynamics...................................... 31 2.4 Relativity.......................................... 33 2.5 Radiation.......................................... 37 2.6 Electromagnetism in Matter................................ 44 3 Statistical Mechanics 52 3.1 Ensembles.......................................... 52 3.2 Thermodynamics...................................... 58 3.3 Entropy and Information................................. 63 3.4 Classical Gases....................................... 66 3.5 Bose{Einstein Statistics.................................. 70 3.6 Fermi{Dirac Statistics................................... 77 3.7 Kinetic Theory....................................... 82 4 Continuum Mechanics 90 4.1 Fluid Statics........................................ 90 4.2 Solid Statics......................................... 94 4.3 Ideal Fluid Flow...................................... 101 4.4 Compressible Flow..................................... 107 4.5 Viscosity........................................... 112 5 Fundamentals of Quantum Mechanics 116 5.1 Physical Postulates..................................... 116 5.2 Wave Mechanics...................................... 120 5.3 The Adiabatic Theorem.................................. 129 5.4 Particles in Electromagnetic Fields............................ 133 5.5 Harmonic Oscillator and Coherent States........................ 138 5.6 The WKB Approximation................................. 144 6 Path Integrals 150 6.1 Formulation......................................... 150 6.2 Gaussian Integrals..................................... 152 6.3 Semiclassical Approximation............................... 154 4 Contents 7 Angular Momentum 159 7.1 Classical Rotations..................................... 159 7.2 Representations of su(2).................................. 161 7.3 Spin and Orbital Angular Momentum.......................... 166 7.4 Central Force Motion.................................... 170 7.5 Addition of Angular Momentum............................. 177 7.6 Tensor Operators...................................... 181 8 Discrete Symmetries 186 8.1 Parity............................................ 186 8.2 Time Reversal........................................ 188 9 Time Independent Perturbation Theory 194 9.1 Formalism.......................................... 194 9.2 The Stark Effect...................................... 197 9.3 Fine Structure....................................... 200 9.4 The Zeeman Effect..................................... 207 9.5 Hyperfine Structure.................................... 211 9.6 The Variational Method.................................. 215 10 Atomic Physics 218 10.1 Identical Particles..................................... 218 10.2 Helium............................................ 221 10.3 The Thomas{Fermi Model................................. 227 10.4 The Hartree{Fock Method................................. 230 10.5 Atomic Structure...................................... 235 10.6 Chemistry.......................................... 239 11 Time Dependent Perturbation Theory 240 11.1 Formalism.......................................... 240 11.2 The Born Approximation................................. 245 11.3 Atoms in Fields....................................... 247 11.4 Quantum Dynamics.................................... 250 12 Scattering 255 12.1 Introduction......................................... 255 12.2 Partial Waves........................................ 257 12.3 Green's Functions..................................... 261 12.4 The Lippmann{Schwinger Equation........................... 267 12.5 The S-Matrix........................................ 270 5 1. Classical Mechanics 1 Classical Mechanics 1.1 Lagrangian Formalism We begin by carefully considering generalized coordinates. • Consider a system with Cartesian coordinates xA. Hamilton's principle states that solutions of the equations of motion are critical points of the action S = R L dt. Then we have the Euler{Lagrange equation d @L @L = dt @x_ @x where we have dropped indices for simplicity. Here, we have L = L(x; x_) and the partial derivative is defined by holding all other arguments of the function constant. • It follows directly from the chain rule that the Euler{Lagrange equations are preserved by any invertible coordinate change, to generalized coordinates qa = qa(xA), because the action is a property of a path and hence is extremized regardless of the coordinates used to describe the path. The ability to use any generalized coordinates we want is a key practical advantage of Lagrangian mechanics over Newtonian mechanics. • It is a little less obvious this holds for time-dependent transformations
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