Someone could believe that Quantum Mechanics has to be presented only in a modern
approach, highlighting from the beginning Principles written in the language of Functional Analysis. Hilbert spaces, Hermitian and unitary operators, eigenvalues and
eigenvectors are part of this modern language from which emerges a frame to understand what we call reality. Ladder operators help us to create and destroy matter. We
jump between energy levels in this way. There are quantum numbers which describe
at a glance the properties of a quantum mechanical system. Some other operators,
which are the generalization of unitary operators in four dimensions, describe both
the polarization of photons and the electron spin. In this modern language, even
the impossibility to predict at the same time the position and the momentum of a
particle is, in fact, an operatorial inequality whose skeleton is the basic CauchyBuniakowski-Schwarz inequality. The atom can be described by the Schrödinger
equation, written in spherical coordinates and solved by some advanced mathematical
technique. This modern language allows analogies with Classical Mechanics considered in its different formulations. For example, starting from the Hamilton-Jacobi
formula in Classical Mechanics, it exists a counterpart in Quantum Mechanics. Only
terms containing è make the difference between classic and quantum formulations.
This advanced language started from Dirac, Heisenberg, Pauli, Ehrenfest,
Schrödinger and all the others who contributed to its development. We believe that
is important to be known by students and it is presented in this book.