Introduction to Quantum Mechanics
Abstract
Quantum Mechanics was born from the realization that Classical Physics could not describe several experiments in which the measurement outcome was discrete rather than continuous as expected. This rendered, among other formulations, Schroedinger’s Equation, a description of this new Physics in terms of Eigenvalue Problems, finding then a natural language in which to model the experimental findings as a set of discrete eigenvalues. The natural space for Quantum Mechanics, given this formulation, is then the one for Eigenvalue Problems in PDE’s, namely Hilbert Spaces, which are (possibly infinite-dimensional) complete vector spaces with an inner product (which allows us to establish angles between vectors). The field of Mathematics related to the study of Hilbert Spaces is Functional Analysis (which is loosely speaking like Linear Algebra for infinite-dimensional spaces). In our case, our points in space (vectors) will be functions, say \(f = f (x)\), \(x ∈ ℝ\) instead of \((f_1, …, f_n ) = (f(x_1 ), …, f (x_n ))\) (the number of “components” of our vectors is not finite anymore, but rather has the cardinality of the real numbers).
References
- Quantum Mechanics in Hilbert Space, by E. Prugovecki, Dover, 2006. Quantum Mechanics in Hilbert Space.
- Morales' lecture notes: introduction to quantum mechanics Notes
- Morales' lecture notes: quantum computing & information (Fall 2022) Notes