The chapter reviews long wavelength mid-infrared quantum cascade lasers (QCLs) emitting between 15 and 28 μm. Historically, 15 μm was a border wavelength above which the QCL performances dramatically degraded, which was partly due to an increase in optical losses in the devices with approaching the Reststrahlen band. This intrinsic limitation caused by multi-phonon absorption sets forbidden or favorable spectral areas depending on the employed materials. The chapter considers specific properties of long wavelength mid-infrared QCLs based on different materials, as well as more general issues related to the QCL design in this long-wavelength frontier of the mid-infrared. The discussed results are presented in the chronological order for each QCL material system, which allows the reader to follow the advances in the field.

The propagation method can be used to describe a particle with wave character moving in an arbitrary one-dimensional potential, $V(x)$. This is done by approximating the potential as a series of potential steps. For a particle of energy $E$ incident from the left, transmission and reflection at the first step is calculated along with phase accumulated propagating to the step and expressed as a $2\times 2$ matrix.

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.

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.

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.

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