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I studied physics as an undergraduate at the University of Texas at Dallas. Below are some of my projects and presentations from 2017-2021.

### Notes & musings

• Bohr Radius from Variational Principle: This is an elegant way to compute the Bohr radius without looking at the Schrodinger equation. It also confirms the virial theorem for a central inverse-square force. This approach was presented to me first in Dr. Yuri Gartstein's PHYS 4301 Quantum Mechanics 1 course.

• Classical Particle in a Cone: the motion of a particle subject to gravity g inside of a cone of opening angle α, solved using the Lagrangian formulation. The equations of motion and numerical solution are shown below.

• Coriolis and Centrifugal Forces in the Plane: the motion in the reference frame of a mass given some initial velocity on a frictionless, flat, surface rotating at ω = 1 rad/s, solved numerically. The Coriolis and centrifugal forces are "fictitious" since they arise from applying Newton's laws to a non-inertial frame; in the inertial frame, the mass will travel in a straight line. The equations of motion and numerical solution for three different initial velocities are shown below.

Since ω points in the z-direction, the above can be reduced to

• Cycloid Motion: the motion of a charged particle subject to perpendicular uniform electric and magnetic fields E = Ez and B = Bx, solved numerically. Amazingly, the particle's average trajectory lies perpendicular to both the electric and magnetic fields.

• Electromagnetic Field and Dual Tensors: I have seen that several electrodynamics texts leave the derivation of the field and dual tensors as an exercise to the reader. I have Lorentz transformed an antisymmetric tensor twice below and have matched the set of transformation rules to previous results from special relativity to arrive at the field and dual tensors.

From our previous knowledge of the Lorentz transformations we know that fields transform as

We see there are six transformations and feel inspired to find an object with six distinct elements—a rank-2 antisymmetric tensor—that matches these results. Such a tensor is of the form

Let's Lorentz transform each of the six distinct elements of the above twice

where the Lorentz transformation matrix Λ is

where γ is the Lorentz factor and β ≡ v/c: So the set of transformations is There are two ways to correspond equations (1)-(6) with (7)-(12) respectively, giving rise to the field and dual tensors.

• Electromagnetism in Tensor Notation: using the above-found field and dual tensors, Maxwell's four equations in differential form become two tensor equations: where is the current density 4-vector.

I will "run the indexes" below, showing that equations (5) and (6) capture the same information as equations (1)-(4). This is often left as an "exercise to the reader" in electrodynamics texts. Running indexes on equation (5):

That is equation (1).

Putting the above three components together gives

which is equation (4).

Now running indexes on equation (6):

which is equation (3).

Putting the above three components together gives

which is equation (2).

In summary, equations (1) & (4) are expressed in (5); equations (2) & (3) are expressed in (6).

• Electromagnetism in the Lorenz Gauge: we reduced Maxwell's four equations to two tensor equations in the above section. We can do better, reducing the two tensor equations to one equation. Recall from electrostatics that Also recall Faraday's law: and that the magnetic field is the curl of the magnetic vector potential: Substituting (3) into (2), Putting together (1) and (4), So we see that the scalar electric potential and the magnetic vector potential entirely describe time-dependent electric and magnetic fields. We can construct a 4-vector: I will verify that the field tensor (derived above) can be written as For reference, also recall that the position four-vector is Running the indexes, So the electric and magnetic fields can indeed be described by (7). Now, substituting (7) into the inhomogeneous Maxwell Equation gives But in the Lorenz gauge, So So (8) becomes Griffiths calls the above the "most elegant formulation of Maxwell's equations." I think it is beautiful.

• Guided Electromagnetic Waves: a loose end from undergrad electrodynamics that I tied up in grad school.

• Numerical Solutions to the Stellar Structure Equations: these solutions are not normalized but can be solved for the boundary conditions of any main-sequence star.

• The Planck quantities: the Planck quantities provide intuition on the various regimes of physics (classical, quantum, relativistic mechanics). This was the starting point of Dr. Xiaoyan Shi's PHYS 4352 Concepts of Modern Physics course.

• Symmetric Top in Gravity: the motion of a symmetric top (I1 = I2 ≠ I3) of mass M (center of mass at height h) with tip fixed in gravity g solved using Euler angles and the Lagrangian formulation, assuming there is no acceleration in the ψ direction. The Lagrangian, equations of motion, and numerical solution are shown below.

• Thoughts on the Maxwell equations: A semi-analytical and semi-reflective document of thoughts on the Maxwell equations.

• Vector calculus theorems: quick, intuitive derivations of the divergence theorem and Stokes' theorem

### Projects & outreach

• Coriolis and Centrifugal Forces: I gave this talk at the Graduate Students of Physics virtual seminar on September 25th, 2020. This was a topic I studied independently over the summer of 2020. It was a rewarding experience, and I was honored to have Dr. Joe Izen, Dr. Xiaoyan Shi, Dr. Amena Khan, Ms. Clara Ann Norman, and my parents in attendance. The abstract can be found here.

• Einstein Solids: this was a group project on Einstein solids presented in Dr. Bing Lv's PHYS 4311 Thermodynamics & Statistical Mechanics course.

• IntelliChoice 2020 Math Camp: I directed this virtual SAT camp for 97 underpriviledged students in the DFW area. I gave daily lectures and wrote my own homework (all now publicly available), covering most of the high school Algebra I, Algebra II, Geometry, and Pre-Calculus curricula. I was fortunate to have several friends volunteer as tutors. The camp (valued at \$297,000) was run free-of-charge, thanks to the support IntelliChoice donors. You can read the 2020 IntelliChoice Annual Report here.

• Literature Review: Radio-Frequency Electromagnetic Field Exposure of Western Honey Bees: I presented a literature review on this physics-related paper published in Nature for HONS 3199 Honeybees & Society in the spring 2021 semester.

• Pappus's Theorem: this was my end-of-semester project for Dr. Mohammad Akbar's MATH 3321 Geometry course, which I took in the summer of 2020. Pappus's theorem belongs to the field of projective, non-metric geometry, which pre-dates coordinate and analytic geometry. It was a fun project that sharpened my presentation and LaTeX skills. It was also nice to work with fellow physics major Tucker Livingston.

• Quantum Electrodynamics: this was my term paper and associated presentation for PHYS 4352 Concepts of Modern Physics.

• Solar Analemma: this was my end-of-semester project for Dr. Phil Anderson's PHYS 3380 Astronomy course, which I took in fall 2018. It is amazing to see just how much the sun's angular position deviates as the seasons change. (I mis-spelled "analemma" throughout the report, though).

• Stellar Spectroscopy: this was my team's final project for Physical Measurements Lab.