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.

It is possible to engineer properties of materials, devices, and systems by changing experimentally available control parameters to optimally approach a specific objective. The following sections demonstrate some potential applications of quantum engineering and show how this may be achieved by the development of efficient physical models combined with optimization algorithms.

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 classical mechanics, the constants of motion of an isolated system are energy, linear momentum, and angular momentum. So far in this book, angular momentum has not been considered. This chapter starts by defining classical angular momentum and then proceeds to find the corresponding quantum operators. Following this, a hydrogenic atom is studied as a prototype application.

The Hamiltonian for $N$ particles subject to mutual two-body interactions

Engineers who design transistors, lasers, and other semiconductor components want to understand and control the cause of resistance to current flow so that they may better optimize device performance. A detailed microscopic understanding of electron motion from one part of a semiconductor to another requires the explicit calculation of electron scattering probability. One would like to know how to predict electron scattering from one state to another – something quantum mechanics can do.

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.

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.