By: Charles D. Kocher
This collection of notes is meant to cover, in brief (that is, within a single summer's worth of effort), all of the physics that departments typically expect of their incoming graduate students. For example, many schools will send out a document of "prerequisites" for first-year graduate courses. The goal of these notes is to cover roughly all of the material that appears on that document in the simplest, but most useful, way possible. These notes are not meant to be a comprehensive review of all of the physics that is covered in an undergraduate education. The philosophy for choosing exactly what to include, and in what way to do it, was as follows: if you, a graduate student, were forced to do one physics derivation completely cold from memory in order to save the world from aliens (or maybe an evil professor), what is the list of possible derivations that I would hope the villains draw from? The material here is the most fundamental core of physics, those topics which are assumed known in almost every discussion you will have with your peers and instructors. I hope you are able to completely internalize everything with the help of these notes.
With that said, the method for deriving results here may not be the same as you saw in your undergraduate courses, causing you some potential difficulties. However, most of the math necessary to understand these notes has been covered in the final Chapter, which you should refer to as necessary. Difficult derivations have been undertaken only where the reward (in terms of generality of the methods/arguments used or of the results) outweighs the inconvenience of the proof.
There are no exercises listed in these notes. The only exercise for each Chapter is for you to go back and try to rederive every result on a blank sheet of paper or a blank whiteboard, and to keep trying until you can do each from memory. As a graduate student, you are likely being paid to do physics now (sorry M.S. students). As someone being paid to do physics, you need to know the core theory of the subject. The objective of these notes is for you, now a professional physicist, to be able to reproduce them. If you succeed in this goal, you will have a strong foundation from which the rest of your career will follow.
Before we begin the required material in earnest, we will first review Newtonian Mechanics in Chapter CM.0. While not strictly necessary, every physicist should have a certain amount of understanding of the beginning of the subject, and it will set up the rest of the material in its proper context. We will then move on to modern classical mechanics. We will cover the basic theory of Lagrangians, equations of motion, symmetries and Noether's theorem, Hamiltonians, and Poisson Brackets, then we will do some of the core applications, like the harmonic oscillator and central potentials.
Next we will study quantum mechanics, beginning with basic naive quantization from the Poisson bracket. We will introduce the Dirac bra-ket notation, eigenvalue problems, change of basis and other linear algebra points, and the Heisenberg and Schrödinger formalisms, as well as the uncertainty principle. We will solve the Schrödinger equation using separation of variables for important problems in one and more continuous dimensions, like the finite and infinite wells. We will also solve it for discrete problems like spins. For the harmonic oscillator, we will introduce raising and lowering operators. Finally, we will cover basic aspects of perturbation theory.
Our third part will be equilibrium statistical mechanics. We will begin with thermodynamics and the definition of entropy. Then we will introduce multiplicities, the Boltzmann distribution, the partition function, and the connections between statistical mechanics and thermodynamics. As examples, we will cover the ideal gas, the equipartition theorem, Einstein and Debye solids, the one-dimensional Ising model, and blackbody radiation. Finally, we will deal with the grand canonical ensemble, degenerate Fermi gases, and Bose-Einstein condensates.
Our final physics section will be on electromagnetism. We will start with special relativity and classical field theories. We can then write down the Lagrangian that leads to Maxwell's equations and the Lorentz force law. From there, we can specify to the usual cases of electrostatics and magnetostatics, where we will solve Poisson's equation, including using the method of images and the multipole expansion. We will also briefly deal with electromagnetism in media, and with circuits. We will then move back to the full theory to discuss electromagnetic waves, optics, and radiation.
The last part of the notes, mathematical methods, is not meant to be read last. It is to be consulted throughout, whenever a problem arises. The first section, vector calculus, is assumed to be known, but we include it anyway. We will cover separation of variables (particularly with the heat/diffusion equation, the wave equation, and for boundary value problems), second order ODEs (including series solutions), and Fourier analysis (both series and transforms). The math is spread throughout the book for the most part, but additional insights and a coherent presentation can be found here.
Each section includes a list of references so that you know where to look for more information if necessary. Best of luck in your learning here and in graduate school!
CM.0.1 Newton's Laws for a Single Particle
CM.0.1.1 Conservation of Momentum
CM.0.1.2 Conservation of Angular Momentum
CM.0.1.3 Conservation of Energy
CM.0.2 Many Particles
CM.0.2.1 Conservation of Momentum
CM.0.2.2 Conservation of Angular Momentum
CM.0.2.3 Conservation of Energy
CM.0.3 References
CM.1.1 Functionals, Lagrangians, and Action
CM.1.1.1 What is a functional?
CM.1.1.2 The Principle of Least Action and the Euler-Lagrange Equations
CM.1.2 Symmetries and Conserved Quantities are Related by Noether's Theorem
CM.1.2.1 Stating and Proving Noether's Theorem
CM.1.2.2 Example Applications of Noether's Theorem
CM.1.3 The Hamiltonian Formulation, Hamilton's Equations, and Poisson Brackets
CM.1.3.1 The Hamiltonian and Hamilton's Equations
CM.1.3.2 Poisson Brackets
CM.1.4 References
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