Quantum Mechanics: Ensembles and Identical Particles
Pure and Mixed Ensemble / State In quantum mechanics, we only talk about average or expectation value for...
Time Evolution of Quantum States
Time-dependent Schrödinger Equation Time-dependent Schrödinger equations is not an eigenvalue equation, but allowing us to predict the state...
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Operations Research (2): Optimization AlgorithmsNational Taiwan University You probably have used some solver / optimizer software to solve...
Quantum Mechanics: Uncertainty Principle and Change of Basis
Commuting Operators Two observables A and B are considered ‘compatible’ if the corresponding operators commute with each other,...
Heuristic Algorithms: A Case Study
The objectives for any research is that we want to build a mathematical model to formulate the given problem. The...
Non-Linear Programming: Gradient Descent and Newton’s Method
Non-Linear Programs When visualizing a linear program, its feasible region looks like a polygon. Because the objective function...
Stern-Gerlach Experiments & Dirac Bra-ket Notation
Electron Spin Electron has intrinsic angular momentum called spin that is not associated with its orbital motion. And...
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Vector Calculus for Engineers The Hong Kong University of Science and Technology We can’t emphasize enough the importance...
Special Relativity: Space and Time
For hundreds years, space and time were absolute in the concept of Newton, they are something that can...
Quantum Mechanics: 1-Dimensional Finite Potential Problems
Finite Potential Well Finite potential well problem is more realistic compared to the infinite potential well. It is...
Time-Independent Schrödinger Equation
Wave-particle Duality The double-slit experiments show us: particle source P1 : probability pattern when slit 1 is openP2...
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Differential Equations for EngineersThe Hong Kong University of Science and Technology By highlighting both theories and applications in...
Vector Calculus: Fundamental Theorems
Gradient Theorem The fundamental theorem of single variable calculus was the one that told you “the integral of the...
Line and Surface Integrals
Line Integrals Scalar Fields We have a curve C in the x-y plane, we can represent a point...
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Logic for EconomistsUniversity of Amsterdam The logo of University of Amsterdam is cool, so is this succinct course....