The composition of subsystems in quantum theory is defined in terms of a mathematical operation known as the tensor product. We proceed to explain this concept, and to show how it fits in the Hilbert space calculus.

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.

The introduction of the Hilbert space as the essential mathematical structure for the formulation of quantum theory was motivated by the following facts.

In Chapter 2, we saw that the nascent quantum theory was divided between two different descriptions of time evolution.

This chapter describes how romantic partners navigate the disagreements that necessarily result from their interdependence and how partners recover when they intentionally or unintentionally hurt each other. Specifically, it reviews the ways in which goals and desires conflict to produce disagreements and how disagreements provide a diagnostic situation in which people make inferences about their partner’s thoughts, feelings, and commitment. Next, it describes typical conflict topics, how conflicts tend to be experienced, and typical conflict prevalence over the course of a romantic relationship. Next, the chapter covers how people manage interpersonal conflicts and highlights specific conflict behaviors that are typically destructive (e.g., hostility, withdrawal) and specific conflict behaviors that are typically constructive (e.g., intimacy, problem solving), as well as how the adaptiveness of conflict behaviors can change depending on the situation. Finally, this chapter reviews how partners can recover from destructive conflicts and other relationship transgressions by accommodating rather than retaliating, sacrificing, and forgiving.

This chapter focuses on how motivated cognition – the tendency for people to think in ways that are consistent with their goals – supports relationship maintenance. It starts with an overview of the assortment of strategies that people use to maintain their close relationships, which lay the foundation for this chapter and subsequent chapters. Then, it explicates the specific biases and illusions that people tend to have about their partners (e.g., seeing partners as particularly physically attractive, projecting ideal traits onto one’s partner) and their relationships (e.g., perceiving one’s own relationship as superior to others). This chapter also reviews empirical evidence describing the implications of this motivated inaccuracy for perceivers and their partners’ relationship experiences. This chapter also describes how biased perceptions extend to potential alternatives, leading people to ignore and devalue attractive alternative partners. It concludes with a discussion of contextual factors that shape the extent to which people engage in motivated cognition and the specific consequences of bias and illusion in relationships.

Given the description of a quantum state in terms of Hilbert space vectors, physical magnitudes (Heisenberg’s matrices) correspond to linear operators on the Hilbert space. A linear operator (or simply, operator) is a linear map of a Hilbert space to itself.

This chapter describes positive interpersonal processes: interactions between people that actively enhance their close relationships. It begins by describing the field’s shift toward studying positive processes and highlighting the utility of considering positive phenomena as unique from negative phenomena. Then, it reviews three interpersonal behaviors that have been shown to enhance relationships and describes the evidence supporting their benefits. First, spending time together (particularly spending time on novel and exciting activities) enhances relationships by enabling partners to meet their self-expansion needs in ongoing relationships. A second positive process is co-experiencing positive emotions, such as joy, amusement, and excitement, which augment and sustain positive experiences and facilitate interpersonal synchrony. Finally, this chapter reviews the benefits of communicating affection and the individual differences in how and how often people express affection.