Quantum mechanics is an essential part of any physics undergraduate or graduate curriculum for its wide range of applicability to different branches of physics. It is taught at different levels, starting from second or third year undergraduate to final year graduate programmes. Through these courses students learn to appreciate the details of the subject at various levels. It is therefore difficult to have a single book which caters to students at all levels. This gap is precisely what we aim to fill here.

The book bore out of several courses on quantum mechanics taught by both the authors at several institutions such as the Saha Institute of Nuclear Physics, the University of Calcutta, and the Indian Association for the Cultivation of Science. These included both undergraduate and graduate level courses. These courses made us realize the necessity of a book which not only provides a clear introduction to the basic tenets of the subject but also presents a thorough discussion of several advanced topics. The latter seemed particularly difficult to find in a single book, which served as a motivation for writing this book.

We have not included a discussion of the old quantum theory in our book. We assumed that the reader would be familiar with that development, which was initiated by Planck and Einstein. That theory deals with particle-like properties of energy. The subject matter of our book is the other side of the wave–particle duality that is the hallmark of modern physics, i.e., the theory which treats matter as waves. The title of this book clearly shows that this theory, `quantum mechanics', will be the sole concern of this book.

Any book on quantum mechanics requires introduction to certain mathematical techniques such as group theory, linear algebra, and differential equations. We provide a somewhat detailed introduction to the first two topics since it is our understanding that these topics may not be taught in detail in other courses that physics students usually encounter before taking their first course on quantum mechanics. However, the last topic is usually well discussed in a generic physics curriculum; we have therefore relegated its discussion to the appendices, making every effort to keep it self-contained.

Silicon integrated circuits (ICs) are pervasive in our world, and the global semiconductor industry today exceeds $500 billion in annual sales. The devices and chips this industry produces support global industries, including consumer electronics, transportation, avionics and many others, that collectively represent a major part of global markets. Devices and chips built with other semiconductor materials such as GaAs, SiC and GaN provide critical components for specific application areas, including high-frequency communications systems, solid-state lighting and power management. It is not incorrect to say that the technical foundation of our modern world is based on semiconductors. The critical role that chips play has led to global competition to design, fabricate and build into advanced systems these remarkable components. Their importance to our world is unlikely to change in the foreseeable future.

Hybrid systems often try to leverage the advantages of one algorithm with the once of another while minimize its own disadvantages. Having discussed fuzzy logic and neural networks as well as a number of optimization algorithms, Chapter 7 presents several hybrid algorithms that can be used for optimization, controls, and modeling. In particular, we look at neural expert systems, expand these systems to neuro fuzzy systems and adaptive neuro-fuzzy inference systems, which we use for control applications. While revisiting the Mamdani and Sugeno fuzzy inference system, the Tsukamoto fuzzy system as well as different partitioning methods are discussed, such as the grid, the tree and the scatter partitioning. Examples using Matlab FIS app as well as Matlab’s ANFIS editor are used throughout the chapter.

In this chapter, we shall discuss a few problems that are not usually addressed for a first course in quantum mechanics. However, these topics are becoming increasingly important for modern-day research in several areas of theoretical physics. We have tried to make the range of topics discussed as broad as possible; however, they are by no means exhaustive.

Coherent states

In this section, we shall discuss coherent states. Such states can be constructed for a wide variety of quantum mechanical systems. We shall, however, restrict our discussion to coherent states of harmonic oscillators, because the concept and analytic treatment of coherent states are particularly simple for harmonic oscillators.

Simple harmonic oscillators were discussed in Chapter 7. The concept of coherent states of such oscillators comes from noting that the annihilation operator a annihilates the ground state: a|0〉 = 0. It can be said that the ground state of the oscillator is an eigenstate of the annihilation operator with eigenvalue equal to zero.

Chapter 3 introduces the concept of the rule base along with the material from Chapter 2 to construct different fuzzy inference systems such as the Mamdani fuzzy inference system or the Sugeno fuzzy inference system. The Takagi-Sugeno fuzzy inference system is used to design fuzzy logic controllers and Lyapunov theory is utilized to investigate the closed-loop system stability of such controllers. Concepts such as local sector nonlinearity, globally asymptotical stability using state-space models are introduced and discussed to fashion controllers for nonlinear systems. Throughout the chapter, Matlab’s FIS editor is used to design fuzzy inference systems and corresponding controllers.

As stated in the concluding sections of Chapter 1, in the light of the interference experiments, we would like to treat the states of a system as vectors. In this chapter, we discuss the general theory of vectors, and how the state vectors fit into that theoretical structure.

Introduction

During the first encounter with vectors, in high school, a student learns that a vector is something that has both magnitude and direction. This definition is unsatisfactory for reasons that we will discuss now. In order to apply this definition, one needs to first define the magnitude of a vector, which is not a vector. The direction is represented by the angles subtended by the vector with some fixed vectors. So we need to develop the notion of the angle between two vectors, and the angle is again not a vector. Alternatively, a vector is described in terms of its components, which are projections of the vector on some fixed axes, or basis vectors. This just means that the definition is incomplete, because now we need to define the basis vectors, i.e., the fixed axes. Rather, a vector should be defined by the fundamental properties of the set of vectors itself.

Let us then look for operations on vectors that produce vectors. We can take some hint from the set of numbers. Given two numbers, we can add them and get a number. We can also multiply them and get a number. There are certain properties that addition and multiplication of two numbers possess, for example, both should be commutative and both should be associative.

In elementary texts on vectors, we certainly learn how to add them, so addition is a property defined on vectors as well. However, multiplication cannot be defined. Although in high school or early college we learn about dot products and cross products of vectors, they do not qualify for our purpose. Dot product does not qualify because the result of dot product of two vectors is not a vector: it is a scalar. The reason cross product does not qualify is that it can be defined only in three dimensional (3D) space, where there can be a unique direction that is perpendicular to two given vectors.

Starting with crisp set theory, fuzzy sets and concepts of fuzzy logic are introduced in Chapter 2. Some of the key operators are discussed and utilized in a number of examples. Membership functions, membership operators, their programming in Matlab, as well as logic operators using membership functions are explained. Along with conditional statements such as fuzzy rules and linguistic variables concepts such as antecedents, consequences and inference are discussed and shown how to implement this type of reasoning in Matlab.

Chapter 1 provides for an introduction to the key concepts of the book, including supervised and unsupervised learning, reinforcement learning, and controls. The objective is to provide an overview of the many methods and algorithms and how they are relate to each other as well as to controls applications.

Based on Chapter 6, in this chapter we expand the discussion of neural networks to include networks that have more than one hidden layer. Common structures such as the convolutional neural network (CNN) or the Long Short-Term Memory network (LSTM) are explained and used along with Matlab’s Deep Network Designer App as well as Matlab script to implement and train such networks. Issues such as the vanishing or exploding gradient, normalization, and training strategies are discussed. Concepts that address overfitting and the vanishing or exploding gradient are introduced, including dropout and regularization. Transfer learning is discussed and showcased using Matlab’s DND App.

In this chapter, we discuss the fabrication of a modern complementary metal-oxide–semiconductor (CMOS) integrated circuit using the individual process steps that are combined in a complete process flow sequence to make the chips. Such an ordered process flow from the sandbox of tools available in different combinations would be used to make any kind of device, such as a biosensor, a microfluidic device or a micro-electromechanical systems (MEMS) device. The wafer’s past history and the future process steps can greatly influence how one chooses to order the individual steps. For example, high-temperature steps at the end of a process could disturb delicate doping profiles introduced early in the process. For this reason, we believe it is worth understanding the choices made in assembling a modern CMOS process flow. Seeing the “big picture” of a complete process flow should also help to put the individual process steps we discuss in subsequent chapters into perspective.

One of the main challenges in designing a front-end process for building a device is accurate control of the placement of the active doping regions. Understanding and controlling diffusion and annealing behavior are essential to obtaining the desired electrical characteristics. Consider a cross-section of a state-of-the-art MOS transistor and imagine what happens when it gets scaled down to smaller dimensions (Figure 7.1). In “ideal” or Dennard scaling, as described in Chapter 1, everything shrinks down linearly from one generation to the next. This means that not only do the lateral dimensions scale, but the vertical dimensions, such as the deep source/drain contacting junctions and the shallower tip or extension junctions, also scale. This maintains the same electric field patterns (assuming the operating voltage also scales proportionally). With the same ℰ-field patterns, the device operates in the same manner as before, except that the shorter channel length allows for faster switching speeds [1].

In this chapter, we establish the mathematical foundation for hard computing optimization algorithms. We look at the classical optimization approaches and extend our discussion to include iterative methods, which hold a special role in machine learning. In particular, we review the gradient decent method, Newton’s method, the conjugate gradient method and the quasi-Newton’s method. Along with the discussion of these optimization methods, implementation using Matlab script as well as considerations for use in neural network training algorithms are provided. Finally, the Levenberg-Marquardt method is introduced, discussed, and implemented in Matlab script to compare its functioning with the other four iterative algorithms introduced in this chapter.