Math and Physics Projects
I am extremely passionate about math and physics. This page houses some projects that I have worked on, including a published research paper. It also includes smaller projects that I have worked on, which have served as great learning opportunities.
(The above chalkboard writing is from my investigation into air resistance)
Quantization of the ModMax Oscillator
Completed October 2023
I co-authored a research paper in the field of theoretical high energy physics titled "Quantization of the ModMax Oscillator" with Dr. Christian Ferko, a postdoctoral researcher at the University of California, Davis. It is submitted to Physical Review D and is expected to be published by the end of 2023. The abstract and Arxiv preprint is below.
We quantize the ModMax oscillator, which is the dimensional reduction of the Modified Maxwell theory to one spacetime dimension. We show that the propagator of the ModMax oscillator satisfies a differential equation related to the Laplace equation in cylindrical coordinates, and we obtain expressions for the classical and quantum partition functions of the theory. To do this, we develop general results for deformations of quantum mechanical theories by functions of conserved charges. We show that canonical quantization and path integral quantization of such deformed theories are equivalent only if one uses the phase space path integral; this gives a precise quantum analogue of the statement that classical deformations of the Lagrangian are equivalent to those of the Hamiltonian.
Desmos Drawing
Completed April 2023
I was selected to attend an event called LASER: Math and Art Day at Montclair State University. I learned about the numerous connections between mathematics and art. I applied what I learned to create an illustration of a Montclair State University building, featuring a ticking clock. I used rotational coordinate transformations to create the numerals on the clock. I also learned the reason why the 4 on many clocks is "IIII" instead of "IV".
Simplistic Physics
Ongoing
As part of an independent study that I completed sophomore year, I created a physics website that explained physics concepts using simple terms and user-friendly diagrams. The goal was to make physics more approachable.
Graphene Rotational Acceleration Particle Hall Sensor (GRAPHS)
Completed April 2023
Nikhil Vijay, William Forte, Rishit Arora, Sahil Shah, Sarvesh Patham and I created a proposal for an experiment to run at CERN. We submitted this to CERN as part of their beamline competition for high school students. This experiment involved testing how graphene responds to particle beams. The response would be measured through a Hall Effect Sensor. This allows us to find things such as movement due to induced radiation pressure, which has applications when creating solar sails.
I am currently not allowed to share the experiment proposal on this webpage. Please contact me if you would like to see it.
Beyond the Second Derivative
Completed January 2023
Given a function's first and second derivatives at a point, we are able to tell if those points are minima or maxima on the original function. However, these tests are inconclusive if we find that the second derivative is equal to 0. It would be possible for a point with zero concavity to be either a minimum, maximum, or neither. Prathamesh Trivedi and I sought to solve this issue by developing a new test that took higher-order derivatives into account, and proved our method was valid by explicitly testing nearby points. We later discovered that there is another way of doing this with Taylor Series, but this is not presented in the project, as we were not aware of Taylor Series at the time.
A Strange Approximation
Completed January 2023
Given two legs of a right triangle with given lengths, it is possible to compute the length of the hypotenuse. However, this involves the Pythagorean Theorem and square roots. These operations tend to be computationally expensive. Prathamesh Trivedi and I present an approximation to this problem using a linear combination of the leg lengths of our triangle. We translate this problem to polar coordinates and prove that our proposed approximation differs from the exact solution by no more than 4%. This is useful for applications where distance needs to be computed extremely quickly but with little precision. An extension to this problem, not presented here, involves using multivariable calculus to derive slightly more optimal constants. Our proposed constants for the linear combination represent the linear combination with the lowest maximum error for two-digit decimals.
Defining the Natural Log
Completed January 2023
In this project, I present a function, defined as a definite integral of the reciprocal function. I proved that this function is equivalent to the natural log function, defined as the inverse of the natural exponential.). I also demonstrate that the presented function has the same properties as the natural log function, such as the product, quotient, and power properties. This was submitted as a project for my AP Calculus BC class. The idea of defining the natural log in this manner is not originally mine.
Snell's Law as an Optimization Problem
Completed January 2023
In this project, Prathamesh Trivedi and I present a derivation of Snell's Law using Fermat's principle. We optimize the pathway that a ray of light takes, given that it will always travel along the trajectory that minimizes the time traveled. We also present an aside that derives this from Huygen's Principle. In addition to being a project for our calculus class, was applied to the NJAAPT Physics Olympics. Our school's physics club team, which we lead, was able to place first in this competition. This was a major factor in allowing us to win first place overall in the entire competition.
Analysis of Euler's Method
Completed January 2023
In this project, Prathamesh Trivedi and I analyze Euler's method. We created a program that is able to take a function, run Euler's method, and then compare it to the solution of the differential equation. This program is in the coding projects section of my site. We find the relationship between the step size and the accuracy of the approximation. Be aware that there was some strange formatting when the document was exported, due to the large tables and images.
Optimizing Roller Coasters
Completed October 2022
In this project, a group of friends and I sought to create a roller coaster that minimized path length, given a set of predetermined boundary conditions. The solution shows how to connect two line segments of different slope with another line, while still being differentiable at all points. In practical application, one could avoid taking the limit and get any level of curvature they want while designing a roller coaster.