According to the Schrödinger equation, a particle with wave character and mass $m$ in the presence of a potential may be described as a state that is a function of space and time. Space and time are assumed to be smooth and continuous. The potential can localize the particle to one region of space forming a bound state.

Scattering experiments are one of our most important tools for extracting information about the structure and interactions of microscopic systems. In these experiments, we prepare a beam of particles of a given type and we direct it towards a target. The interaction of the particles in the beam with those of the target may lead to various phenomena: changes in the direction and the energy of incoming particles, absorption of incoming particles, the appearance of new species of particles, and so on. The target is surrounded by particle detectors that identify the particles that exit the interaction region and measure their momenta.

In previous chapters, we saw that quantum theory is unique among physical theories, in that its predictions refer exclusively to measurement outcomes rather than to the properties of physical objects. It is therefore no surprise that the study of quantum measurements has developed into a research field on its own. The earlier studies of quantum measurements focused on conceptual and foundational issues, but in recent years quantum measurement theory has become a crucial tool for quantum technologies.

Quantum mechanics is a very successful description of atomic scale systems. The mathematical formalism relies on the algebra of noncommuting linear Hermitian operators. Postulates provide a logical framework with which to make contact with the results of experimental measurements.

Quantum mechanics is a basis for understanding physical phenomena on an atomic scale. An electron point particle of rest mass $m_0$, charge magnitude $e$, and quantized spin magnitude $\hslash /2$, can behave as a wave.

In previous chapters, we encountered the fundamental principles of quantum theory and we saw how the representation of physical magnitudes by Hilbert space operators allows us to construct probabilities for the outcome of any experiment. However, these principles do not tell us what the fundamental physical systems are, how to construct their associated Hilbert spaces, and which operators correspond to physical magnitudes.

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 $|0〉$.