In this section, we will analyze transitions between energy eigenstates caused by a transient external force, for example, an EM pulse. We will assume that the external force is weak so that we can describe these decays by a version of perturbation theory. This results into very simple expressions for the transition probabilities and rates with universal validity, that is, they apply to all kinds of phenomena from atomic to nuclear and high-energy physics.

The simplest quantum systems correspond to the smallest nontrivial Hilbert space ${ℂ}^2$. They are called two-level systems. Some physical magnitudes, such as photon polarization or electron spin, are naturally described by the Hilbert space ${ℂ}^2$. However, usually, two-level systems are approximations to more complex systems. Consider, for example, an atom with a lowest-energy state $|0〉$.

Many experiments require the execution of several measurements in a microscopic system. For example, consider a particle A decaying into a pair of particles B and C that move in different directions. Each product particle is detected by a different apparatus at different moments of time. We need a rule that tells us how to incorporate information obtained from the first measurement in order to predict the outcome of the second.

One of the most important motives for the development of quantum theory was the need to understand the structure of atoms and molecules, and to account for their emission spectra. Later on, analogous issues were raised for other composite systems, such as nuclei and hadrons. In all cases, the answer requires finding the discrete spectrum of the Hamiltonian $\widehat{H}$ that describes the composite system, that is, solving the eigenvalue equation $\widehat{H}|n〉=E_n|n〉$ for the Hamiltonian.

In previous chapters, we saw in detail the immense success of quantum theory in describing microscopic systems. We also saw that it leads to concrete predictions that are persistently being confirmed by experiment, even predictions that grossly violate all physical intuition that had been generated by 250 years of classical physics.

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.

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.

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.