Newtonian mechanics was the first great synthesis of modern physics. It provided the main theory about the workings of the physical world from the date of its first presentation (1687) until the beginnings of the twentieth century.

The study of composite systems typically requires their analysis into simpler systems. In classical physics, the simplest systems are particles, that is, pointlike bodies that move in space. A particle is traditionally described by three position coordinates and three momenta, so the associated state space is ${ℝ}^6$. In Chapter 14, we will see that this description is incomplete: Particles also have an additional degree of freedom called spin.

In classical mechanics, a particle of mass $m$ subject to a restoring force linear in displacement, $x$, from a potential minimum such that $F(x)=-{\kappa }_{0}x$, where ${\kappa }_0$ is the force constant, results in one-dimensional simple harmonic motion with an oscillation frequency $\omega =\sqrt{{\kappa }_0/m}$.

The fundamental kinematical symmetry is the invariance under transformations between inertial reference frames. In the regime of small velocities, this symmetry corresponds to the Galilei group. However, the Galilei symmetry is only approximate. The exact symmetry, in the absence of gravity, is defined by the Poincaré group, which we analyse here.

In Chapter 11, we saw that by Wigner’s theorem, symmetry transformations in quantum theory are represented by unitary or antiunitary operators. Then, we focused exclusively in the symmetry of space rotations. Since our focus was so narrow, we did not have to introduce the most appropriate language for the description of symmetries, namely group theory.

In Chapter 9, we presented the quantum rule of combination of subsystems through the tensor product. In this chapter, we will discuss a key elaboration of this rule that applies to composite systems with a specific symmetry, namely, invariance under exchange of identical particles.

Spin was introduced as part of the effort to understand the structure of atoms prior to the development of mature quantum theory. Variations of Bohr’s model described atoms in terms of three quantum numbers, roughly similar to $n,𝓁$, and $m$ that appear when solving the Schrödinger equation in central potentials. In this context, Pauli proposed that, in each atom, there exists at most one electron for each triplet of quantum numbers. This proposal is Pauli’s famous “exclusion principle,” which we will analyze in Chapter 15.

Often there are situations in which the solutions to the time-independent Schrödinger equation are known for a particular potential but not for a similar but different potential. Time-independent perturbation theory provides a means of finding approximate solutions using an expansion in the known eigenfunctions.

The history of the laser dates back to at least 1951 and an idea of Townes. He wanted to use ammonia molecules to amplify microwave radiation. Townes and two students completed a prototype device in late 1953 and gave it the name maser or microwave amplification by stimulated emission of radiation.

The first step towards quantum theory was a response to a problem that could not be addressed by the concepts and methods of classical physics: the radiation from black bodies.

One of the most important concepts of classical mechanics is that of a closed system. A closed system is loosely defined as a system whose components interact only with each other, and it is characterized by phase space volume conservation and energy conservation – see Section 1.2.

We saw in Chapter 2 that Born’s statistical interpretation of the wave function was one of the building blocks of quantum theory. According to Born’s interpretation, the wave function of a particle at a given moment of time defined a probability density $|\psi (x){|}^2$ with respect to the position $x$. This result is generalized to state vectors of an Hilbert space and to general observables through the following procedure.