This page collects links to various short notes I've made in the last few years on various mathematical topics. Mostly they're intended as useful reference for me, or to help clarify my thoughts. If anyone else finds them useful, that's fantastic!
The Pauli Matrices and the Bloch Sphere. A rigorous development of the basic formulas involving the Pauli matrices, and the Bloch sphere.
Notes on the Schrödinger Equation. A derivation of the 1-dimensional Schrödinger equation.
Asymptotic Decay of Bound Solutions to the Schrödinger Equation A rigorous proof that sufficiently regular bound solutions to the time-independent 1-dimensional Schrödinger equation either blow up (unphysically) or else have exponential decay.
Heisenberg's Inequality. Derivation of Heisenberg's inequality for hermitian operators (observables) on Hilbert space.
The trace, partial traces, and entanglement A rigorous development of partial traces and some basics facts on their applications to the theory of entanglement. Also a rigorous definition of the Dirac bra and ket notation.
Quantum Phase Estimation. Mathematical proof of the algorithm using quantum Fourier transformation to perform phase estimation.
Phase and Group Velocity. A quick derivation of classical group velocity of a wave packet, with applications to free particle solutions of time-indep SE.
"30min to QC". A Colab using Cirq to give a very quick intro to both core QM ideas of superposition and entanglement, followed by a demo of the Deutch-Joza Algorithm.
Quantum Teleportation A Colab using Cirq to demonstrate and motivate a slightly simplified verion of quantum teleportation.
Grover's Algorithm. An implementation of Grover's Algorithm in Cirq.
Quaternions, SU(2), and SO(3). A development of the basic algebraic properties of the quaternions and their relationship to the Lie groups \(SU(2)\) and \(SO(3)\).
Lagrangian and Hamiltonian Mechanics. A very brief derivation of the equations of motion in Lagrangian and Hamiltonian machanics. The main example is the derivation of the Hamiltonian of a charged particle moving in an EM field, and its quantization.
Fourier Notes. A place to collect precise statements of various useful facts from Fourier Analysis.
The Cauchy Interlacing Theorem. Notes and proofs of the Cauchy Interlacing Theorem, which relates the eigenvalues of a submatruces to the eigenvalues of the whole matrix. Also notes on applications to singular value theory.
Unbounded Operators. Short development of the most basic facts about unbounded operators on Hilbert space.
Unitary Equivalance of Nest Algebras Short (and incomplete) development of the theory of unitary Equivalance of nests and nest algebras.
The Halting Problem. Prove the Halting Problem is algorithmically undecidable.
Left Ideals. Notes on left ideals of rings and the Jacobson radical.
Matching Theorems. (Incomplete) notes on the Gale-Shapley Stable Marriage Theorem.
Bibliography. BibTeX file used in this site and in my recent papers (rendered nicely in HTML, though).