Mathematical Foundations
Our first section on quantum mechanics will introduce the fundamental concepts of quantum mechanics. In this chapter, we will introduce the state vector formalism, which is the most fundamental concept in quantum mechanics.
Through this chapter, we shall gradually build up the mathematical framework of quantum mechanics.
At the same time, we will slowly develop our notation to a more abstract and concise form.
This means, for example, we will eventually strip away the arrow on top of a vector:
📄️ Hilbert Spaces and Dirac Notation
Our first section on quantum mechanics will introduce the fundamental concepts of quantum mechanics.
📄️ Observables and Operators
In quantum mechanics, we are interested in the properties of particles, such as their position, momentum, energy, and so on.
📄️ Commutators and Uncertainty
Uncertainty is one of the most puzzling, yet most fundamental, aspects of quantum mechanics.
📄️ Matrix Representations of Operators
In the previous sections, we introduced the state vector formalism of quantum mechanics, as well as some important results derived from it.
📄️ Change of Basis
There are many cases in quantum mechanics where we need to change the basis of a vector from one basis to another.
📄️ Applying to Spin 1/2 Systems
In the previous sections, we introduced the main concepts of the state vector formalism in quantum mechanics.
📄️ Position and Momentum
In the previous section, we applied the state vector formalism to spin-1/2 particles.
📄️ Wave Functions in X-P Space
In this note, we will discuss the wavefunctions in the position-momentum space. We will see how the wave functions in the position space and the momentum space are related to each other.
📄️ Summary
In the previous sections, we have introduced the mathematical foundations of quantum mechanics.