What is international law, and how is it different from domestic (national) law?  This chapter provides an answer to those questions and introduces other foundational concepts in international law such as opinio juris, sovereignty, and the state. We trace the development of international law from Western/European perspectives and discuss how international law was often experienced as oppression and violence in the non-Western world. The chapter concludes with a consideration of the classic sources of international law: customs, treaties, general principles of law, judicial decisions, and scholarly writings, each of which is presented in some detail.

How can the use of force by states be constrained under international law?  Under what circumstances has the use of force by states been deemed "legitimate"?  How are rules about the legitimate use of force changing?  These questions are examined in depth, along with the relevant instruments of international law.  The chapter details the various ways that "force" and "aggression" have been defined and used.  It treats all the major forms of force or coercion, including full-scale military operations, economic sanctions and reprisals, proxy and clandestine forces, small-scale conflicts.  It then traces growing prohibitions on the use of force from the League, the UN, the ICJ, and other sources.  Finally, the Laws of Armed Conflict are highlighted as are the Geneva Conventions and instruments limiting munitions and conflict on the land, air, and sea.  The use of drones and autonomous weapons systems, guided by AI, are highlighted as a growing area of concern for international law.

What role do international organizations play in international law? Similar to states, they have international legal personality, responsibilities, and immunities.  This chapter focuses on the preeminent global intergovernmental organization, the United Nations, and details the functions and limits of its principal organs.  Special attention is given to the General Assembly, Security Council, and International Court of Justice. The European Union is the leading example of a regional, supranational organization, and its authority and institutions are discussed in detail as well. The chapter concludes with brief considerations of other major international organizations, including the North Atlantic Treaty Organization, the Organization of American States, the African Union, and the World Health Organization.

In this chapter, we will introduce the , which is a formulation of quantum mechanics equivalent to the Schrödinger equation, but has a profoundly distinct interpretation. Further, the path integral is very easily extended to incorporate special relativity, which is very challenging and inconvenient within the context of the Schrödinger equation. So, what is the idea of this path integral? Our goal will be to calculate the amplitude for a quantum mechanical particle that starts at position xi at time t = 0 and ends at position xf at some later time $t=T>0$. In some sense, this question is analogous to what you ask in an introduction to kinematics in introductory physics; however, its analysis in quantum mechanics will prove to be a bit more complicated than that in the first week of your first course in physics.

The mathematical and physical construction of quantum mechanics is undeniably beautiful, but as hinted in the Introduction, why the universe should ultimately be quantum mechanical is mysterious. Of course, empirical science can never answer the question of “Why?” definitively, but only establish the rules that govern Nature through experiment. Nevertheless, there were several points in our discussion of the motivation for the Hilbert space, the Born rule, or the Dirac–von Neumann axioms that seemed to be completely inexplicable and potentially inconsistent with the guiding principles we used. In this chapter, we survey a few of these points from an introductory, modern perspective. Quantum mechanics works, makes precise predictions, and agrees with experiment, but what quantum mechanics is is still very much an open question.

We’ve come a long way from the fundamental mathematical properties of linear operators to the profound physical interpretations of them. We have finally developed the necessary background for deriving the fundamental equation of quantum mechanics, the Schrödinger equation.

How have states coordinated their response to protecting the environment? This chapter outlines the major features of international environmental law, beginning with a brief history of global environmental governance.  We begin with a broad outline of the major environmental conferences, culminating in the architecture of climate change governance. While climate change is a focus of this chapter, we also highlight some of the other major environmental achievements related to protecting the marine environment, regulating atmospheric pollutants, and preserving biodiversity.  Core environmental law principles are detailed and illustrated with prominent cases for clarity.  The final sections examine the intersections between human rights, armed conflict, and the environment, concluding with a discussion of the impact of corporations on the global environment and the responsibilities they might bear for restoring it.

How do we define the state in international law, and what is its relationship to individuals? We begin by outlining the state as a legal concept and differentiating it from similar concepts. We then explore the legal personality of the state under international law, including the elements of statehood, absolute and restrictive immunity, and state responsibility. The problems of state recognition (or non-recognition) of other states and governments is a key to understanding how states interact, as are changes in state status (e.g., secession or other consequential changes).  The last half of the chapter is devoted to the reciprocal responsibilities state and individuals have toward one another, focusing on nationality, citizenship, refugees, statelessness, and the state’s treatment of foreign nationals.

In the previous chapter, we studied the consequences of rotations and angular momentum in three spatial dimensions, building up to the topic of this chapter: the quantum mechanics of the hydrogen atom. Hydrogen is, of course, the lightest element of the periodic table and consists of a proton and an electron bound through electromagnetism. Our goal for studying this problem is to determine the bound-state energy eigenstates, just like we did with the infinite square well and harmonic oscillator. These energy eigenstates will then tell us how the proton and electron are positioned with respect to one another in space, as well as the energy levels and how energy is transferred when the hydrogen atom transforms from one energy level to another. As always, our goal is to diagonalize the Hamiltonian $\hat{H}$; that is, determine the states $|\psi ⟩$ and energies E such that