Lithography is arguably the most important process step in modern integrated circuit (IC) manufacturing. The ability to print patterns with features as small as 10–20 nm and to place those patterns on a substrate with a precision of a few nanometers is what makes today’s chips possible. Virtually all ICs are manufactured today with deep-ultraviolet (DUV) optical lithography operating with 193 nm photons, the basic process introduced in Figure 1.7.

In this chapter, we shall treat quantum mechanical problems where the system Hamiltonian either has explicit time dependence or has some parameter that is time-dependent. In either case, the Schrödinger equation is time-dependent and we are going to discuss techniques, either perturbative or exact, to solve such equations. In §12.1, we shall discuss the general formalism for treating such problems. Next, in §§ 12.2 and 12.3, we provide an analysis of the timedependent Hamiltonian for a two-level system. This will be followed by §12.4, where perturbative solutions to time-dependent Schrödinger equation shall be outlined. Finally, we analyze some aspects of Hamiltonians with periodic time dependence in §12.5.

In this chapter, we introduce some of the more popular ML algorithms. Our objective is to provide the basic concepts and main ideas, how to utilize these algorithms using Matlab, and offer some examples. In particular, we discuss essential concepts in feature engineering and how to apply them in Matlab. Support vector machines (SVM), K-nearest neighbor (KNN), linear regression, Naïve Bayes algorithm, and decision trees are introduced and the fundamental underlying mathematics is explained while using Matlab’s corresponding Apps to implement each of these algorithms. A special section on reinforcement learning is included, detailing the key concepts and basic mechanism of this third ML category. In particular, we showcase how to implement reinforcement learning in Matlab as well as make use of some of the Python libraries available online and show how to use reinforcement learning for controller design.

Since a great many of the examples of quantum systems described in this book involve solutions of second-order differential equations, in this appendix we lay down the basic techniques for finding these solutions. We do not care to be completely general. We only discuss the kind of equations that we encounter for the purpose of this book.

Symmetries play a very important role in quantum mechanics, much more so than in classical mechanics. The reason for this is the vector space structure of the state vectors of a quantum system, as will be clear in the discussion of this chapter.

Symmetry and conservation

Our intuitive notion of symmetry is through geometrical objects. A square is more symmetric than a rectangle, whereas a circle is more symmetric than both. We can make the notion quantitative if we consider what are the operations on these geometrical objects that produce a result that is indistinguishable from the original one. For a rectangle, rotations in its plane by 180 or its multiples produce an identical figure. For a square, rotation by 90 and its multiples do the job, and this is why a square is more symmetric than a rectangle. For a circle, any rotation in its plane leaves the shape unchanged, and that is why the circle is more symmetric than either the square or the rectangle.

We can extend this notion to mathematical expressions as well. For example, suppose we are considering a particle in a 1D space, with a potential that obeys the condition

In this case, we can say that the potential is symmetric under the transformation that changes the sign of x, something that we will denote by x→ −x. From the Heisenberg equation of the system, Eq. (4.4, p. 83), we see that this transformation on x would also imply that p changes sign: p → −p. Since the kinetic energy term in the Hamiltonian is quadratic in p, it does not change under this transformation. Thus, the entire Hamiltonian is invariant under the transformation if the potential obeys Eq. (5.1).

Let us now discuss what happens if we apply a transformation to all vectors in a vector space and find that the physical consequences of the transformed vectors are indistinguishable from those of the original ones. In that case, we would call the transformation a symmetry transformation.

Starting with the perceptron, in Chapter 6 we discuss the functioning, the training, and the use of neural networks. For the different neural network structures, the corresponding script in Matlab is provided and the limitations of the different neural network architectures are discussed. A detailed discussion and the underlying mathematical concept of the Backpropagation learning algorithm is accompanied with simple examples as well as sophisticated implementations using Matlab. Chapter 6 also includesconsiderations on quality measures of trained neural networks, such as the accuracy, recall, specificity, precision, prevalence, and some of the derived quantities such as the F-score and the receiver operating characteristic plot. We also look at the overfitting problem and how to handle it during the neural network training process.

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.