Chapter 22 builds on its preceding chapter, discussing “security detention,” confinement without criminal charges for security reasons during armed conflicts or other situations of violence. This is presumably the fate of most, if not all, of the forty remaining Guantánamo detainees. The chapter compares security detention with other forms of wartime internment, including World War II Swiss internment of US and German aircrews. Law, LOAC or otherwise, authorizing internment during non-international armed conflicts is sparse but authorities (such as this textbook) point to a 2014 UK case, and the criticism that followed its resolution, as pointing the way to LOAC (Sections III and IV of 1949 Geneva Convention IV) that would allow and control security detention. Much of the chapter describes present-day US security detention policy. A difficulty is squaring LOAC with security detention beyond the conclusion of hostilities. Surprisingly, however, the US Supreme Court has a lon, if not unanimous, record of approving security detention well past the end of hostilities. Those cases suggest that current Guantánamo detainees are probably going to age and die in place.

We are now ready to study a generic class of three-dimensional physical systems. They are the systems that evolve in a central potential, i.e. a potential energy that depends only on the distance 𝑟 from the origin.

Chapter 14 describes lawful human targets, the first being enemy combatants. Their required characteristics, activities, and locations are covered, followed by exceptions to their lawful targeting: prisoners, those showing intent to surrender, and those incapacitated by wounds. Wrongful use of force investigations are detailed, to impress the fact that violations of an exempt targeting status have consequences within our own armed forces. What legal standard is applied to the shooter when investigating possible unlawful use of force? Lawful civilian targets are considered, as well: when one is targeted he may return fire on civilians in self-defense; and when civilians are directly participating in hostilities (DPH), as is often the case in recent conflicts. Significant space is devoted to explaining what constitutes DPH, as well as to continuous combat function, the higher form of DPH. Targeting of heads of state (e.g., Saddam Hussein, Adolf Hitler, the US president) is covered. Then, is there a duty to capture, rather than wound? To wound, rather than kill? Finally, the politically fraught killings of Iranian General Suliemoni and Anwar al-Aulaqi are examined for LOAC compliance.

We shall now look at the solutions of Schrödinger’s equation for the quantum harmonic oscillator. In this chapter we will focus on the one-dimensional case, which can be seen as another example of a one-dimensional potential. Unlike the infinite potential well, the potential for the harmonic oscillator is finite for all finite values of 𝑥, and only diverges when 𝑥 → ±∞.

This chapter is concerned with the relationship between frequency domain operations (using the DFT from ) and time domain operations (using discrete transient convolution from ). Frequency and time domain operations are connected by a convolution theorem, showing that multiplication in one domain corresponds to convolution in the other. This provides guidance to avoid unwanted wrap-around effects when performing DFT linear filtering. It also motivates the use of window functions as multipliers in both the time and frequency domains to improve linear filtering performance and DFT spectral estimates, as described in .

Symmetries play a central role in the study of physical systems. They dictate the choice of the dynamical variables used to characterise the system, lead to conservation laws and set constraints on the evolution of the system. We shall see explicit examples of these features in this chapter, and set up the mathematical framework to be able to have a unified formalism for describing generic symmetries.

Inwe have introduced the basic concepts of quantum mechanics, and studied them for some simple, yet relevant, one-dimensional systems. In this chapter we take another step towards the description of real physical phenomena and generalise the concepts introduced so far to systems that evolve in more than one spatial dimension. The generalisation is straightforward and it will give us the opportunity to review some of the key ideas about physical states, observables and time evolution. In the process, we will encounter and highlight new features that were not present for one-dimensional systems. Once again let us emphasise that three-dimensional in this context refers to the dimension of the physical space in which the system is defined, and not to the dimensionality of the Hilbert space of states; the latter clearly will depend on the type of system that we consider. The three-dimensional formulation will allow us to discuss more realistic examples of physical systems. It will be clear as we progress through this chapter that everything we discuss can be generalised to an arbitrary number of dimensions. In some physical applications, where a quantum system is confined to a plane, a two-dimensional formulation will be useful. More generally, it is instructive to think about problems in arbitrary numbers of dimensions. In this respect, it is fundamental to be able to work with vectors, tensors, indices, and all that. Problems and examples in this chapter should help develop some confidence in using an index notation to deal with linear algebra.