Quantitative Analysis
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Differential equations are one of the most important mathematical tools in theoretical physics. In my research, I used differential equations to understand the time evolution of a qubit state. The normal evolution of qubits involves a simple solution to the Schrodinger Equation. When there is a driving force that is resonate with the energy spacing of the two states of a qubit, then the solution of the Shrodinger Equation is a bit more complex, but still uses the standard techniques of calculus. The measurement of solid-state qubits involves forces coming from stochastic noise that requires the techniques of Ito calculus. Each stage of evolution must be analyzed separately using the appropriate techniques.
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Bayesian Analysis
Bayes' Theorem is a very famous and fundamental theorem in probability theory. Simply put, it describes quantitatively how to adjust one's beliefs when new evidence is considered. For instance, it is more likely that a person with long hair is a woman because it is more likely that women have long hair in our culture. In simple logic, if A implies B, then one cannot assume that B implies A. It is important to remember that conditional probabilities are statistical correlations, which are expressed in similar terms to logic; however, these cannot be interchanged when making logical arguments. Implying, correlating, and causing are extremely different verbs; remember, correlation does not imply causation.
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Combinatorics
Combinatorics is an especially elegant system for counting possibilities. I used combinatorics as a tool to calculate the frustrated magnetic ground state of electrons on a Kagome lattice. Very simply, a complete basis for the state space of N electrons can be found by counting all pairwise connections of N points spread evenly around a circle (shown to the left).
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