This website focuses on using linear algebra to describe the properties of quantum states. Relying on a foundation of quantum principles and linear algebra concepts, we can use Matlab to model different quantum states to better understand their time evolution and other properties. This website serves as a repository of key quantum concepts as well as programs which allow for an observation of the time evolution of quantum states, both stationary and non-stationary, in the position basis and the energy basis. Other programs serve to demonstrate the probability density of finding the particle at a specific position as well as calculate the expectation value for position and energy for a certain energy state. The relationship between position and momentum can be better understood through the use of a Gaussian wavepacket which fits the probability density to a specific location. Selection rules for electronic and vibrational transitions have also been further developed through modeling and visualization in Matlab. Ultimately, simplifying assumptions are explored in order to determine the molecular dynamics of chemical bonding. These key quantum concepts, and others, are developed and explained on different supporting pages, and the Matlab code used to demonstrate these concepts are also linked below.

1. Summary of Key Quantum Ideas
2. Particle in a Box - 1D
3. The Time Dependent Schrodinger Equation
   • Change of Basis
   • Stationary States
   • Non-Stationary States
   • Gaussian Wavepackets
4. Differential vs. Difference Equations
5. Introduction to Pertubation Theory
6. Selection Rules
   • Simple Harmonic Oscillator
   • Electronic Transitions
   • Morse Potential
7. Fourier Transform
8. Born-Oppenheimer Approximation
9. Molecular Dynamics

• 3D Plot Code
• Eigsort Function
• Particle in a Box
• Stationary States
• Non-Stationary States
• Gaussian Wavepacket
• Position and Momentum with Gaussian Wavepackets
• Selection Rules
• Morse Potential
• Pertubation Theory