We have finally developed all the tools that are necessary to study the hydrogen atom (H atom) from a quantum-mechanical perspective. In this chapter we present a non-relativistic formulation of the problem, where the interaction is modelled with a static Coulomb potential.

One-dimensional systems are an ideal playground to test the concepts introduced so far in this book. Solving a few examples explicitly and discussing their physical interpretation will allow us to get familiar with the description of quantum systems and their dynamics. In this chapter we will solve Schrödinger’s equation for some simple one-dimensional potentials. Having found the solution of the mathematical equations, we will focus on how to extract physical information from these solutions. For these problems it is convenient to work in the position representation and describe the state of the system by its wave function.

One of the most common methods for studying the structure of matter and fundamental particles is to scatter particles and measure the outcome of the collision. For the collision between two particles, the problem is posed such that initially both particles are widely separated and moving towards each other but are assumed to evolve independently of any influence from each other. Then the two particles come within close proximity and interact with each other, resulting in their initial state being altered. Subsequently, the two particles in their altered form separate and some distance away this altered state is measured.

Chapter 18 attempts to make cultural property attractive to LOAC students. Destruction and pillage of cultural property have been a wartime problem for hundreds of years. The 1954 Hague Convention protecting cultural property is one of the more significant treaties combatants should be aware of, regardless of the blandness of the topic to many warfighters, because it is a potential fast-track to trial. The 1954 Convention and its first Protocol (of two) were issued on the same day, reflecting ineptitude in their drafting. Nevertheless, the Convention’s pertinent articles and its sometimes-unique terminology are discussed. It is the second Protocol (1999) that gives clarity and precision to the Convention, which is closely examined. Several fictional scenarios are employed to make clear the Convention’s workings. Its relationship to the 1977 Additional Protocols is strong, as is shown. The Cases and Materials examine two prosecutions arising from the conflict in the former Yugoslavia that demonstrate that the Convention has teeth that bite combatants who disregard its prohibitions.

We now examine problems where the Hamiltonian depends on time. For such cases, energy is not conserved and so there are no stationary states. In general, for such problems it is very difficult to find exact solutions to the Schrödinger equation. Where progress can be made is in situations where time dependence is only in a small part of the Hamiltonian. Time-dependent perturbation theory is the formal approach to address such problems.

Chapter 16 shows torture to be a lose-lose practice, its results virtually always unreliable. It is shown to be contrary to customary international law, the domestic and military laws of virtually every state, as well as the Geneva Conventions, the Rome Statute of the International Criminal Court and, of course, LOAC. Yet torture persists. After describing torture, international examples are detailed, including from CIA black sites. Excuses for torture are given and rebutted. Particular attention is given to US uses of torture after 9/11, none of which has been shown to have been useful, contrary to presidential assertions. Israel’s torture experience is covered, as is the French experience in Algeria. Waterboarding is clearly identified as torture, with several examples provided. The “ticking time bomb” scenario is argued to be an intellectual fraud, despite its being a frequent excuse for both liberals and conservatives to promote it. US military practice is also detailed. Cases and Materials include a real-world Sri Lankan ticking time bomb experience and a copy of a chilling US Department of Justice memorandum describing – and authorizing! – CIA black site methods of torture.

Chapter 21 addresses military commissions. Their history and evolution are described. During the Mexican-American War, they were a modest military law stopgap in foreign countries. The US Civil War and the Lieber Code (Chapter 2) made them what they are today: rough-edged military field tribunals to address the war crimes of foreign combatants. Their value was proved following World War II, when all the Allied states employed them. Guantánamo’s post-9/11 perverted version of military commissions, in the shadow of CIA black sites and “enhanced interrogations,” may have ended their utility: initiated by civilian political appointees with the aim of denying prisoners not just POW status but basic human rights and legal protections, Guantánamo’s military commissions have been an abject failure, brought to heel by civilian courts’ legal resistance and forced recognition of civilian legal oversight. Guantánamo’s record of eight convictions in eighteen years, three of which have been vacated, is proof of their legal ineffectiveness and moral hollowness.

Chapter 7 discusses LOAC’s core principles. This text holds that there are four core principles, per the International Court of Justice. Distinction requires that Parties to a conflict always distinguish between civilians and civilian objects and combatants, and fire only on military objectives. Military necessity, the second principle, is very elastic, holding that all lawful measures to defeat the enemy quickly and efficiently are allowed in LOAC. The third principle, unnecessary suffering, directs that one may not employ weapons or projectiles that cause superfluous injuries or unnecessary suffering in enemy fighters. This principle is directed only to combatants, not civilians who should not be targeted at all. Finally, the principle of proportionality directs that civilian injury or loss of life, and damage or destruction of civilian objects, may not be excessive in relation to the direct and concrete military advantage to be gained. The problem with the four principles is definitional: what will “quickly” defeat the enemy? What injuries are “superfluous”? What incidental civilian death is “excessive” to a military advantage? Such answers as there are, are in this chapter.

Chapter 6 relates the individual status of the “players” on an armed conflict battlefield. It is the second question students should answer (after the conflict’s status) because the players’ status determines their rights and legal duties in the conflict. Civilians and combatants predominate, of course, but there are numerous subcategories for both: prisoners of war, retainees, militia, persons accompanying the armed forces, levée en masse fighters, spies, and mercenaries. Each category is explained and placed in relation to the other players. What if a civilian captured with weapons claims noncombatant status? LOAC provides for an informal tribunal. How is the familiar “farmer-by-day-fighter-by-night” dealt with? Who is an “unprivileged belligerent” and are they the same as an “unlawful combatant”? What status for a civilian who directly participates in hostilities and what are their battlefield rights, if any? Who is a “protected person” and what makes them such? These statuses and more are considered, along with their positions vis-à-vis the combatants who engage in combat with them.

Chapter 8 puts previous chapters to work by defining war crimes – their required precursors and circumstances – asking the reader whether real-life scenarios are, or are not, war crimes. The import of grave breaches is illustrated, including the factual World War II murder of fifty prisoners of war by Nazis, the subject of the motion picture, The Great Escape. When can a mosque be fired on, if ever? Are suicide attacks violations of LOAC? Is hostage-taking ever lawful? What is pillage? And “double-tapping”? On the battlefield, may one feign death? What if an individual volunteers to be a human shield? Are human shields ever lawful? Real-world war crime cases and their outcomes are related. “Disciplinary” war crimes, seldom noted, are, here, as well. Continuing, universal jurisdiction for war crimes turns out to be less than universal, and civilians can commit war crimes just as combatants can. Rape and other gender crimes are covered, here, as well. Four war crime trial records are included in the chapter’s Cases and Materials.

Having described the states and the observables of a quantum system, we shall now introduce the rules that determine their time evolution.

Now that we have developed the tools to describe three-dimensional systems, we are ready to introduce into our quantum-mechanical framework the concept of angular momentum. Recall that in classical mechanics angular momentum is defined as the vector product of position and momentum.