An introduction to solid state physics. Logistics Heat Capacity of Solids Einstein Model of Heat Capacity of Solids Debye Model of Heat Capacity of Solids Electrical Properties (Drude Model) Constant E&M Force Thermal Conductivity Sommerfield Model Fermi-Dirac Statistics Sommerfield Model (cont.) LCAO (Linear Combination of Atomic Orbitals)/ Tight Binding Theory Chemistry Review Shell Theory Ionic Bonds Covalent Bonds LCAO Basics Vibrations 2 Types of Springs Crystal Structure Reciprocal Space Logistics Location: 307 Pupin Time: 2:40-3:55 PM Tuesdays and Thursdays Textbook: The Oxford Solid State Basics Grading Scheme 30% problem sets (5 in total....

An introduction to plasma physics. Logistics Grading Scheme MHD Equillibrium Consequences of MHD Equillibrium Theta Pinch Z Pinch Screw Pinch Toroidal Geometry Definitions Forces Grad Shafranov Equation Stream Functions Grad Shafranov Derivation Sketch Aspect Ratio Expansion Logistics Location: Mudd 825 Time: 1:10-2:40 Book: MHD Stability of Tokamaks by Zohm Grading Scheme Projects: 60% Homework: 40% MHD Equillibrium Suprisingly useful since particles in plasma undergo gyration, which enhances number of collisions and causes discrete particle to behave more fluid-like Setting time derivatives in MHD equations to zero This is a good assumption, since deviations from this result in motion with timescales on the order of microseconds (called the Alfven time $\tau_{A}$) Hence, if you are physically observe the plasma, good chance it is in EQ $\vec{v} = 0$ is the magneto-static limit (or that $v« v_{a}$) where $v_{a}$ is the Alfven speed (which in turn is defined as size of detector divided by $\tau_{a}$) In space, the magneto-static approximation is normally not valid Define the plasma beta as $\beta = \frac{p}{\frac{B^{2}}{2\mu}}$ or the thermal energy divided by the magnetic field energy Equation of state $P = n_{e}T_{e}+n_{i}T_{i}$ Momentum equation: $\rho(\frac{\partial}{\partial t}\vec{v}+\vec{v}\nabla\cdot \vec{v}) = \vec{J}\times B =\nabla P$ Maxwell’s equations $\nabla \times \vec{B} = \mu_{0}\vec{J}$ $\nabla \cdot B = 0$ $v \times B +E =0 \rightarrow E = 0$ $\frac{\partial }{\partial t} B = -\nabla \times E \rightarrow \frac{\partial }{\partial t} B =0$ Continuity $\frac{\partial}{\partial t}\rho + \nabla \cdot v = 0$ 0=0 Energy $\frac{\partial}{\partial t}P+\vec{v}\cdot\nabla p -c_{s}^{2}(\frac{\partial}{\partial t}\rho + \vec{v}\cdot \nabla \rho) = 0$ To Summarize, for MHD equilibrium, we have that...

Logistics Conventions Newtonian Gravity The Two Body Problem Kepler’s Second Law Kepler’s Third Law From Kepler To Newton The Equivalence Principle Gravitational Redshift Light falls Differential Geometry Definitions/Basics Crash Course Example 2-Sphere Special Relativity Minkowski Spacetime Diagrams Metric Useful Tricks 4-Vectors 4-velocity Dual Vectors 4-momentum Inverse Metric Variational Approach Moving to GR Christoffel Symbols We haven’t done GR yet. Let’s do that Qausi-Stationary Curved or not Curved Tensors General Covariance Useful facts Scalar Field Transformation Vector Field Transformation Dual Vector Transformation Stress-Energy Tensor $T^{\mu\nu}$ Dust Covariant Derivatives Properties of Covariant Derivatives Tidal Forces of Curvature Newtonian GR version Directional Covariant Derivative Reimannian tensor Reimannian Tensor properties Parallel Transport Stress Energy Tensor Conservation EM Force Law Einstein’s Equation What the Heck is $\kappa$?...

Ohm’s Law Drude Model Flux Maxwell’s Equations Mutual Inductance Self Inductance Energy in Fields Maxwell’s Equation in Matter Electric Magnetic Boundary Conditions Continuity Equations Charge Energy Momentum Angular Momentum Waves Basic Waves Polarization EM Waves Reflection and Transmission Absorption and Dispersion Absorption Dispersion Wave Guides Potentials Gauge Freedom Coulomb Gauge Lorenz gauge Leinard-Weichart Potentials Radiation Dipoles Point Charges Ohm’s Law $\vec{J} = \sigma (\vec{E}+v\times \vec{B}) \approx \sigma \vec{E}$ $J$ is the current density $\sigma$ is the conductivity $\vec{E}$ is the electric field Microscopic description typically, the velocities are so slow that we can drop the magnetic field term $\vec{I} = \int \vec{J}\cdot \vec{dA}$...

Grading Breakdown Heat Equation Steady state solutions Time Dependent Solution Seperation of Variables Laplace Equation Polar Coordinates Mean Value Property Maximum Principle Fourier Series Convergence Term by Term Differentiation Solving Inhomogeneous Equations Wave Equation Strum Liouville Theory SL properties Green’s Identity Minimization Principle Parseval’s Identity Bessel’s Identity Multi-Dimensional PDEs Green’s Identity (Higher Dimensions) Polar Coordinates Bessel’s equation Fourier Transforms Green’s Functions Heat Equation Case Study Dirac Delta Green’s Function via Differential Equation Nonhomogeneous BCs Green’s Function via Eigenvalue Expansion Fredholm Alternative Grading Breakdown HW: 40% Assigned on Monday weekly....

Basic CLI commands Vim Cheatsheet Standard Library Functions Compiling and Linking Gcc Header Files Lecture 2 Make Git Lecture 3 Data Types in C Sizeof() Computer Representation of Numbers Signed versus Unsigned Operator Gotchas Precedence Left shift Short Circuit Pre and Post Increment Undefined Behavior Expressions vs Statements Storage Classes Automatic Variables (Local) Static Variables Memory Address Space Stack Data Pointers Void Pointers Faux Pass by Reference NULL pointers Pointer Errata Dangling Pointers Arrays Pointer Arithmetic GUT of Pointers and Arrays Heap Heap Safety Strings String Literals argv Constant Pointers Function Pointers Parsing Function Pointers Structs Unions Typdef Libraries IO Standard I/O Stdout Stdin Stderr Redirection Pipes Files Blocking and Reading Buffering File Seeking Formatted IO Inspecting binary files Endianness Network byte order Forks Reaping Children exec() UNIX Users and Groups Shell Scripts TCP/IP netcat File Descriptors Sockets API Connect Bind, Listen, Accept Send and Receive HTTP/1....

Notes for Mechanics class for Spring 2023. Chapter 1 Main Concepts Scalars, Vectors and Matrices Derivatives of Position Chapter 2 Main Concepts Newton’s Laws Definitions Conservation Laws Chapter 3 Main Concepts Chapter 4 Main Concepts Nonlinear Oscillations Phase diagrams Plane Pendulum Chapter 5 Main Concepts Chapter 6 Main Concepts Chapter 7 Main Concepts Chapter 8 Main Concepts Chapter 9 Main Concepts Chapter 10 Main Concepts Chapter 11 Main Concepts Euler Equations Force Field Chapter 1 Main Concepts Scalars, Vectors and Matrices A scalar is some quantity that is invariant under any coordinate transformation A vector is an object in $\mathbb{R^{n}}$ that is invariant under rotations Vectors are closed under addition and scalar multiplication The Dot product is defined as $\vec{A}\cdot \vec{B} = \Sigma_{i} A_{i}B_{i}$ or equivalently $\vec{A}\cdot \vec{B} = |A| |B| \cos(\vec{A},\vec{B})$ $|A| = \sqrt{\Sigma_{i} A_{i}^{2}}$ This is invariant under rotation (since it is a scalar) Abelian The Cross Product is defined as $C = A \times B$ where $C_{i} = \Sigma_{j,k} \epsilon_{ijk}A_{j}B_{k}$ whre $\epsilon_{ijk}$ is the Levi-Civata tensor (0 if indices are the same, 1 if an even permutation of 1,2,3 and -1 if an odd permutation of 1,2,3) Alternatively $|C| = |A||B|\sin\theta$ Non-Abelian $A\times(B\times C) = (A\cdot C)B-(A\cdot B)C$ $A\cdot (B\times C) = B\cdot (C\times A) = C \cdot (A\times B) = ABC$ Matrices are objects in $\mathbb{R^{n\times m}}$ A simple 2D rotation is given by (can be derived by rotating the primed coordinate system around the orign by some angle and adding up lengths) $x_1^{’} = x_1 \cos \theta +x_2 \sin \theta$ $x_2^{’} = -x_1 \sin \theta +x_2 \cos \theta$ In general $x_{i} = \Sigma_{j=1}^{3} \lambda_{ji} x_{j}^{’}$ where the $\lambda$ are the directional cosines defined by $\lambda_{ij} = \cos(x_{i}^{’},x_j)$ ie....

Compilation of notes for Fundamentals of Computer Systems class for Spring 2023. Problem Solving Tips Administrative Stuff Exams Grades Lecture 0 Lecture 1 CPU Models Addition in different bases Definitions Modular Arithmetic Integer format with word size restriction Unsigned binary numbers Binary Addition Algorithm (BAA) Negative numbers Signed Magnitude Representation 1’s Complement Problems with Signed Magnitude and 1’s Complement 2’s complement Intuition behind 2’s complement Lecture 2 Why 2’s complement? Easy subtraction Detecting Overflow Floating Points IEEE standard for a 32 bit word Doubles Underflow Boolean Algebra Boolean Algebra Identities Distributive Law Proof DeMorgan’s Theorem Consensus Theorem circuit Representation of Boolean Algebra Coverting Circuits to Booleans NAND and NOR XOR Duals Lecture 3 Sum of Products (SoP) Product of Sums (PoS) Convert SoP to PoS Minterms Maxterms Karnaugh Map K-map Notation Summary of Simplification with k-maps 2-bit multiplier Don’t Care Conditions Drawing Circuits Lecture 4 Standard Circuits Enabler Decoder Circuit Decoder With Enable MUX (Multiplexer) Representing Functions with Decoders and MUXes MUX trick Shifter Circuit Barrel Shift left w/ Wraparound L-R Shift Circuit with Rollout Unsigned Adder Circuit Half-Adder Full-Adder Signed Adder/Subtractor (2’s-C) Ripple Carry Adder Optimizing Ripple Carry (Carry Lookahead) Merge with known 0’s Code Converter Contraction Example 1 Lecture 5 Latch Intuition SR Latch SR Latch with Control D Latch with control Latches Can’t be Clocked Flip Flops JK Flip Flop (JKFF) Flip Flop Table Trigger Types Sequential Circuit State Machines Registers Register MUXing Shift Register Ripple Counter PLA (Programmable Logic Devices) MIPS Programming Parts of a Program Computer Hardware CPU Memory Clock ISA (Instruction Set Architecture) Programming in MIPS Instruction Types Memory Pointers $pc sp Constants Sign-extend Pseudoinstructions Things to Remember Assembly Code ALU Details of Memory Single Memory Cell Coincident Selection Extending Memory Why is memory not clocked?...

Compilation of notes for Quantum I class for Spring 2023. The Basics The Current Density Operator (probability current) Erenhfest’s Principle Uncertainty Princple Time Independent SE Interference of Stationary States Solving TISE $V(x)=V_{0}$ Infinite Square Well Properties of Eigenfunctions Finite Square Well Dirac Delta Function Free Particle Fourier Transform Definition Scattering States SHO Commutation relation of $\hat{x}$ and $\hat{p}$ Ladder Operators Hermetian Operators Momentum Eigenvalues Fourier transforms Generalized Uncertainty Relationship Dirac Notation Solution to Spherically Symmetric Schrodinger’s Equation Angular Momentum Spherical Coordinates Spin Electron in Magnetic Field Addition of Angular Momenta The Basics The Wavefunction Written in 1 spatial dimension as $\Psi (x,t)$....

Compilation of notes for Machine Learning class for Spring 2023. Administrative Stuff Topic 1 Workflow of Classification Problems (Supervised Learning) Statistical Approach Classifier Maximum Likelihood Estimation Example Convergence Naive Bayes Classifier How do you quantify the quality of a classifier? Approaches to Classification Generative Approach Discriminative Approach Topic 2 Nearest K-Neighbor Importance of Closeness for k-Nearest Neighbors Distances Similarities Issues with the nearest neighbor k-NN Optimality Proof Practical Considerations of k-nearest neighbor Finding the k-th nearest neighbors takes time Metric of Closeness is Sometimes Unclear Derivative w....