THE FIRST LAW OF THERMODYNAMICS can be mathematically expressed in a variety of ways. All these expressions, however, are easily viewed as rearrangements of the statement that energy can neither be created nor destroyed but only converted from one form to another. In contrast, there is no single universally agreed statement of the second law of thermodynamics. Kline [1] indicates that many seemingly different statements have been accepted as the second law, all of which, however, can be shown to be equivalent after careful and sometimes subtle application of logic. This multiplicity of apparently disparate statements can lead to confusion in understanding the second law.

IN THIS CHAPTER, we see how steady-flow devices combine to form complex systems for power production, propulsion, heating, and cooling. Not only are such systems important from an engineering perspective, but more generally, they are essential to everyday life in industrialized societies. Here we analyze these systems to understand their basic operation and to determine various performance measures. Thermodynamic cycle efficiency, first introduced in Chapter 6, provides a dominant theme for this chapter. Here we investigate in some detail the energy conversion efficiency of the various systems just listed. In some sense, this chapter is the culmination of all the preceding chapters. Chapter 9 not only provides an opportunity to integrate knowledge gained from previous chapters but also provides interesting applications that can be explored in parallel with earlier chapters.illustrates the key role that Chapter 9 plays in our study. The system analysis is shown as bridging, or using, all the previous topics.

IN THIS CHAPTER, we review the concept of energy and the various ways in which closed and open systems can possess energy at both microscopic (molecular) and macroscopic levels. We also carefully define heat and work, which are boundary interactions and, therefore, not properties of a system or control volume. The chapter concludes with a brief examination of the rate laws that govern heat transfer.illustrates how this chapter relates to other thermodynamics topics. We begin with a brief historical overview of our subject matter.

THE FIRST LAW of thermodynamics can be mathematically expressed in a variety of ways. All these expressions, however, are easily viewed as rearrangements of the statement that energy can neither be created nor destroyed but only converted from one form to another. In contrast, there is no single universally agreed upon statement of the second law of thermodynamics. Kline [1] indicates that many seemingly different statements have been accepted as the second law, all of which, however, can be shown to be equivalent after a careful and sometimes subtle application of logic. This multiplicity of apparently disparate statements can lead to confusion in understanding the second law. In this chapter, we examine several statements of the second law and discuss the consequences of each.

IN THIS CHAPTER, we apply the basic conservation principles and other key concepts to analyze a number of important devices. Here we investigate typical components of more complex systems; these components include nozzles, diffusers, throttles, pumps, compressors, fans, turbines, and heat exchangers. In Chapter 9, we will combine these simple devices in more complex systems, which include steam power plants, jet engines, other power and propulsion cycles, heat pumps, refrigeration cycles, and air conditioning and humidification systems.

IN THIS CHAPTER, we apply the fundamental principle of energy conservation to both closed and open thermodynamic systems. In our analyses of closed (fixed-mass) systems, we will express energy conservation for incremental and finite changes in state and at an instant. In dealing with open systems, we again follow a hierarchical development by starting with simple, steady-state, steady-flow cases and then adding detail and complexity to arrive at more general statements of energy conservation. We apply the energy conservation principle to analyses of steady-flow devices, linking our theoretical developments to practical applications.illustrates the key role that this chapter plays in our study.

There's a great deal of interesting physics in the Schrödinger equation and its solutions, and the mathematical underpinnings of that equation can be expressed in several ways. It's been my experience that students find it helpful to see a combination of Erwin Schrödinger's wave mechanics approach and the matrix mechanics approach of Werner Heisenberg, as well as Paul Dirac’s bra and ket notation. So these first two chapters provide the mathematical foundations that will help you understand these different perspectives and “languages” of quantum mechanics, beginning with the basics of vectors in Section 1.1. With that basis in place, you can move on to Dirac notation in Section 1.2 and abstract vectors and functions in Section 1.3. The rules pertaining to complex numbers, vectors, and functions are reviewed in Section 1.4, followed by an explanation of orthogonal functions in Section 1.5, and using the inner product to find components in Section 1.6. The final section of this chapter (as in all later chapters) is a set of problems that will allow you to exercise your understanding of the concepts and mathematical techniques presented in this chapter. Remember that you can find full, interactive solutions to every problem on the book's website.

And since it's easy to lose sight of the architectural plan of an elaborate structure when you’re laying the foundation, as mentioned in the Preface you’ll find in each section a plain-language statement of the main ideas of that section as well as a short paragraph explaining the relevance of that development to the Schrödinger equation and quantum mechanics.

As you look through this chapter, don't forget that this book is modular, so if you have a good understanding of the included topics and their relevance to quantum mechanics, you should feel free to skip over this chapter and jump into the discussions of operators and eigenfunctions in Chapter 2. And if you’re already up to speed on those topics, the Schrödinger equation and quantum wavefunctions await your attention in later chapters.

Vector Basics

If you pick up any book about quantum mechanics, you’re sure to find lots of discussion about wavefunctions and the solutions to the Schrödinger equation. But the language used to describe those functions, and the mathematical techniques used to analyze them, are rooted in the world of vectors.

The concepts and techniques discussed in the previous chapter are intended to prepare you to cross the bridge between the mathematics of vectors and functions and the expected results of measurements of quantum observables such as position, momentum, and energy. In quantum mechanics, every physical observable is associated with a linear “operator” that can be used to determine possible measurement outcomes and their probabilities for a given quantum state.

This chapter begins with an introduction to operators, eigenvectors, and eigenfunctions in Section 2.1, followed by an explanation of the use of Dirac notation with operators in Section 2.2. Hermitian operators and their importance are discussed in Section 2.3, and projection operators are introduced in Section 2.4. The calculation of expectation values is the subject of Section 2.5, and as in every chapter, you’ll find a series of problems to test your understanding in the final section.

Operators, Eigenvectors, and Eigenfunctions

If you’ve heard the phrase “quantum operator” and you’re wondering “What exactly is an operator?,” you’ll be happy to learn that an operator is simply an instruction to perform a certain process on a number, vector, or function. You’ve undoubtedly seen operators before, although you may not have called them that. But you know that the symbol “ √ ” is an instruction to take the square root of whatever appears under the roof of the symbol, and “d( )/dx” tells you to take the first derivative with respect to x of whatever appears inside the parentheses.

The operators you’ll encounter in quantum mechanics are called “linear” because applying them to a sum of vectors or functions gives the same result as applying them to the individual vectors or functions and then summing the results.