 ## Introduction

We live in a Euclidean world, where we reference most things in everyday life to space (x, y, z) and time (t) coordinates. Using Einstein’s theory of special relativity, we will consider space and time together as a 4-vector in order to understand the strange behaviors of space-time events in two different reference frames. In another context, it turns out to be very useful to use conjugate “Fourier transformed” coordinates, the three components of the ‘wave vector’ (Kx, Ky, Kz) and energy E, a description which is incredibly powerful as a mathematical tool in solving equations that arise in physics, chemistry, engineering. This cluster will use both of these views of Space-Time to investigate applications to electricity and magnetism, optics, and the quantum behavior of solids.

### Core Courses

#### Fourier Methods to Compute Energy Bands of Solids

We will see how viewing the world in Fourier space can be used to learn about the ‘band structure’ of electrons in a solid, which describes the energy levels and their momentum dependence. This viewpoint gives great insight into the behavior of real materials, such as semiconductors that underlie all of modern electronics and computers. Without assuming any prior experience, we will introduce Python as a tool in computational physics. We will use Python’s capabilities to do linear algebra and compute energy bands of solids by ‘diagonalizing’ appropriate matrices and investigating phenomena such as Van Hove singularities, Dirac points, flat bands, and more!

#### Special Relativity, Electromagnetism, and Optics

The Lorentz transformation equations describe the relationships of the space-time 4-vector in two reference frames moving with relative constant velocity near the speed of light. The same equations describe the relationships of the momentum-energy 4-vector and show the origin of Einstein’s famous equation E=mc^2. We will discover the phenomena known as length contraction (moving meter sticks look shorter) and time dilation (moving clocks run slower). Then we can understand the famous paradoxes based on traveling twins and the pole in the barn. We will then use special relativity to see that electricity and magnetism are inextricably tied together. Pictures of electric field lines related to accelerating charges explain the origins of radiation fields associated with electromagnetic waves, like light. This will lead us to examine the behavior of light waves in optics. We will study optical phenomena relating to reflection, refraction, mirrors, lenses, polarization, interference, and diffraction.

#### Electronics and Optics Experiments

Each student will receive a set of electronics and optics parts to do home experiments and build projects. In addition to learning about basic dc and ac circuits, students will program an Arduino microcontroller in C++ to get input from sensors (switches, thermometers, etc.) and control output to devices (LEDs, motors, etc.). We will use the Arduino to build a simple oscilloscope to look at electrical voltage signals as a function of time, followed by a fast Fourier transform (FFT) to see the frequency components of the signal. Using a microphone as the source for the input signal, we will analyze the frequencies in musical sounds.

Each student will build an individual electronics project. Examples of possible projects are a robot with wheels and sensors, a controllable lamp with varying colors, an alarm system, a temperature controller, a computer game, and a water irrigation system. We will also use a set of simple components (laser diode, polarizers, slits, diffraction grating, curved mirrors, lenses, photodiode with amplifier) to investigate elementary optics phenomena. We will use the lenses to build a simple microscope and telescope. The final optics project is building a spectrometer to look at the frequency spectrum of different light sources, such as fluorescent bulbs, LEDs, and light from the sun showing Fraunhofer absorption lines.