In this chapter we will discuss the , for which the potential is simply 0 for all $x\in (-\mathrm{\infty },\mathrm{\infty })$. That is, the Hamiltonian is just the kinetic energy

In this chapter, we will study the quantumwhich, just as in classical mechanics, will be our “canonical” quantum system and one which we can apply to a huge variety of systems. Unlike the infinite square well, we will find that the harmonic oscillator is not most naturally expressed in position space. This new formulation of the harmonic oscillator’s Hamiltonian will demonstrate the power of exploiting commutation relations in quantum mechanics and will be a foundation for analyses in future chapters. Eigenstates of the harmonic oscillator, especially its lowest-energy state, have a deep connection to the Heisenberg uncertainty principle and open the door to studying a whole class of states that bridge the chasm between quantum wavefunctions and classical particles.

How do states create agreements with one another, and how do they ensure they are followed? The first half of this chapter details the process by which states make, maintain, and rescind treaties. We elaborate on the treaty process set forth in the Vienna Convention on the Law of Treaties, and explain different treaty types (e.g., bilateral vs. multilateral, self-executing vs. non-self-executing). Reservations—portions of a treaty a state does not wish to join—are also discussed, as well as their ramifications. The second half of the chapter establishes the centrality of diplomats to the creation and execution of much of international law, including treaties.  We elaborate on diplomatic and consular functions as well as diplomatic immunity and asylum.  Diplomats are noted as change agents in the international system, but the potential for abuse of that immunity is always present.

What is the relationship between the global economy and international law? In this chapter, we examine instruments that reflect the liberalism that has prevailed in international trading relations for the last half-century.  The resulting instruments include the articles of the World Trade Organization, the World Intellectual Property Organization, and the Trade-Related Aspects of Intellectual Property Agreement. We also highlight anti-corruption instruments and various non-governmental organizations that also share the goals and processes of international liberalism.   The middle section examines attempts to combat the various and increasingly sophisticated forms of corruption in international business transactions, especially the explosion of difficult-to-combat cyber fraud.  The latter part of the chapter notes a growing trend towards economic nationalist goals, and anti-competitive behavior among state and business elites.

How does international law protect human rights?  We trace the development of human rights, focusing on the international response to the atrocities of World War II and the rapid pace of human rights conventions.  We demonstrate how the signing of the Universal Declaration of Human Rights started a wave of other conventions designed to codify the treatment to which every human is entitled.  Not every state can achieve the wide array of protections these documents outline, but the UN has established methods for reporting violations that provide some minor satisfaction. The last half of the chapter is devoted to a discussion of regional human rights mechanisms, focusing on the European, Inter-American, and African systems and noting their jurisdictional differences. Finally, the development of mechanisms to respond to genocide, war crimes, and crimes against humanity are detailed, including the criminal tribunals set up for specific atrocities and the International Criminal Court.

Our study of linear algebra in the previous chapter suggested some interesting features of a deeper structure. First, as a study of the derivative operator suggested, we will consider a complex vector space, spanned by some set of orthonormal vectors $\{{\vec{v}}_i\}$, where orthonormality is defined by

In this chapter, we are going to step back from the exact solutions of the Schrödinger equation (i.e., diagonalization of the Hamiltonian) and introduce methods for approximation. We have nearly exhausted those problems for which there exists a known, exact, analytic solution to the Schrödinger equation, and much of modern research on quantum mechanics and its generalizations focuses around approaches to approximating the eigenvalues of the Hamiltonian. We will just barely scratch the surface of techniques for approximation in this chapter, focusing on four different methods. The techniques we introduce here are general techniques for determining approximations to eigenvalues of linear operators or matrices, but applied to the problem of solving a quantum system, namely, diagonalization of its Hamiltonian.

How far can a state extend its jurisdiction beyond its borders? In this chapter we make the distinction between state sovereignty and jurisdiction and then outline the five principles of extraterritorial jurisdiction. By providing examples for each jurisdictional principle, the differences between them are sharpened and the ways in which they may overlap are clarified. An interesting area for extraterritorial jurisdiction is the Internet; a brief examination of the European Union’s General Data Privacy Regulation serves to highlight the complexities. The chapter concludes with two practices states can use to attempt to pull alleged criminals who are in another state back into their own jurisdiction for prosecution.  Extradition is a legal process agreed to by states. Rendition, though also used by some states, is not and runs afoul of human rights.

Up until this point, we have essentially exclusively studied the quantum mechanics of individual particles. In this chapter, we will consider ensembles of quantum mechanical particles and attempt to understand their collective dynamics (or at least provide a brief introduction). By “ensemble,” I mean that we are considering a quantum system with more than one particle in it. So, instead of, say, one spin-1/2 particle, we will consider a collection of N spin-1/2 particles, or N particles in the infinite square well, or something like that. With this set-up, we can immediately see that the quantum description of states on a Hilbert space is insufficient and incomplete for describing this collection of particles.

What are the physical limits of state sovereignty? This chapter reviews international law concerning the territorial sovereignty of states, including on the land, the oceans, and the air and space above.  Specific legal instruments that govern these areas are examined in some depth, including the Chicago Air Convention of 1944 and the United Nations Convention on the Law of the Sea (UNCLOS).  Six methods of obtaining sovereign territory are discussed along with vivid examples of each.  The struggle to allow fair use by states and companies over the resources in and below the high seas and outer space is highlighted.  The new space race may prevent the original space treaties’ vision of a new frontier that is devoid of commercialization and militarization.

With scattering and its formalism in the previous chapter, we pivot to studying quantum mechanics in more than one dimension. In this and the next chapter, we will work to generalize our formulation of quantum mechanics to be more realistic (i.e., actually account for the multiple spatial dimensions that we experience in our universe). In this chapter, we will introduce a profound consequence of living in multiple spatial dimensions and a study of angular momentum in quantum mechanics. We will start by setting the stage for describing rotations and later, construct the complete theory of angular momentum, at least as much as we need here.