Notation and Representation in Quantum Mechanics

Quantum mechanics (QM) uses some unique notations to present its mathematical formalism. Due to historical reasons along the development path of QM, different scientists also used different representations in QM and its application. Even today, different notations and representations are still mixed in modern QM research and applications. These caused lots of confusions for beginners. Apparently, we have felt this confusion in the class on Tuesday. For example, I prefer using Dirac notation, i.e., the vectors (ket or bra) to represent the state of a physical system, however, a Schrödinger wavefunction (as a function of the coordinates) is used when we try to solve the eigenvalue equation for a hydrogen atom. What is the relationship between the Dirac state vector and the Schrödinger wavefunction? To help understand these notation and representations, I put this short note together and hope to give you a clear “picture” about how to understand QM’s mathematical formalism.