This paper shows a numerical study of the time-independent Schrödinger equation in one dimension using finite difference methods. We analyze bound states in infinite and finite square wells and explore quantum tunneling through potential barriers. We include a full Python implementation and generate labeled figures illustrating wavefunctions and energy levels for each case. The results offer a visually and quantitatively rich understanding of quantum confinement and tunneling.

This paper explores the Buddhist doctrine of anattā (no-self) and how it fundamentally challenges Western philosophical understandings of identity. By comparing these views, I examine how the idea of no-self impacts metaphysical, psychological, and ethical understandings of identity in the West, and suggests ways the two traditions can inform one another.

CRISPR-Cas9 is a revolutionary tool that allows scientists to edit DNA with precision. This paper explores how CRISPR-Cas9 functions in E. coli, how to design effective guide RNAs, and how to avoid and detect off-target effects. We discuss various experimental approaches, analyze editing efficiency, and outline future directions for improving CRISPR-based bacterial genome editing.

This paper investigates the relationship between minimum wage increases and teenage employment in the United States using state-level panel data from 2010 to 2020. Using visual analysis, regression techniques, and supplemental research, we find a modest negative correlation between minimum wage hikes and teen employment rates in California and New York, while Texas, with a constant minimum wage, showed relatively stable teen employment. We also consider alternative explanations and policy implications, while acknowledging data limitations and suggesting directions for future research.

This research paper explores the interplay between fractals and chaos in physical systems, showcasing how complex, self-similar structures emerge from simple nonlinear dynamics. Using both analytical and computational approaches, we investigate systems such as the logistic map, the Mandelbrot set, and the Lorenz attractor.

The Rubik’s Cube is a mathematical object hiding layers of symmetry and structure. In this paper, we explore how group theory provides a powerful framework for understanding the Rubik’s Cube. We introduce the concept of a group, define the Rubik’s Cube group, and discuss key ideas such as permutations, generators, subgroup structure, and group actions. We aim to show how accessible, tangible puzzles can reveal profound mathematical insights.

In this paper, I explore Descartes’ method of doubt, its philosophical foundations and implications, the pivotal insight of Cogito, ergo sum, as well as the challenges and legacies of his ideas in both classical and contemporary thought. Through this exploration, we see that Descartes' quest for certainty continues to shape how we think about truth, knowledge, and belief.

The goal of this experiment is to measure and understand charge quantization. We will be repeating Robert Millikan's famous oil drop experiment, at a smaller scale, so as to further this goal of observing charge quantization. By carefully following Robert Millikan's oil drop experiment of observing rise and fall times of charged oil drops, we compare the results of our own data to that of the a larger dataset, which differ in quantity by multiple folds.

This paper presents an empirical analysis comparing the volatility of Bitcoin with two traditional financial assets—gold and the S&P 500 index—over a five-year period from 2020 to 2024. By computing daily returns and rolling annualized volatility, we demonstrate that Bitcoin exhibits significantly higher and more erratic volatility than either gold or the S&P 500. We further examine how macroeconomic events such as the COVID-19 pandemic, regulatory shifts, and market cycles contribute to volatility.

This paper explains how spectral graph theory can be used to measure and improve the resilience of networks—how well they can withstand failures or attacks. A key focus is on the second smallest eigenvalue of the Laplacian matrix and how it can be used to assess a network's robustness. We also look at different types of networks—random, scale-free, and small-world—and use simulations to show how spectral properties change when parts of the network are removed.