We have discussed 1D problems in Chapter 7. In this chapter, we discuss mainly 3D problems, although some of the generalities might apply to spaces whose dimensions are difierent.

Generalities

The stationary state form of the Schrödinger equation for a single particle system was given in Eq. (4.24, p. 86). For non-relativistic systems with velocity-independent potential energies, the general form of the Hamiltonian would be

for a particle of mass M. Accordingly, the Schrӧdinger equation for stationary states has the form

The Laplacian operator, ▿2, involves second-order derivatives with respect to the coordinates. This is the difierential equation we want to solve in this chapter for various choices of the potential.

Particle in a 3D box

The problem of a particle in an infinite rectangular-shaped 3D potential well is just an extension of the same problem in 1D. So, the solution can be borrowed from what we have already done in §7.2, but there will be some new features of the solutions which need some attention.

The potential is infinite except within the box defined by

Inside the box, the potential vanishes. The wavefunctions for the stationary states can be written in the form

and the difierential equation corresponding to each factor will be the same as the corresponding 1D problem.

The interface between a semiconductor and an insulator often determines the viability of the material combination in device structures. Silicon is unique in nature, at least among the semiconductors, for having a robust, reliable oxide that can be grown on its surface. The interface between Si and SiO2 is perhaps the most carefully studied of all material interfaces, and is probably the principal reason why silicon has been the dominant semiconductor material. The fact is that silicon naturally oxidizes in the sense that it can be simply placed in a furnace at high temperature with oxygen or water vapor and one obtains a nice, stable dielectric material that is essentially electrically perfect. This distinguishes silicon from all the other simple column IV semiconductor materials. Germanium can be oxidized, but its oxide is soluble in water, which makes it very hard to do any sort of chemical processing.

We have laid down the basic techniques for solving second-order homogeneous differential equations. In this appendix, we apply these techniques to some differential equations of special interest, which appeared at various parts of the text. We perform series solutions to identify the functions that emerge as solutions of these equations. Note that for some of these differential equations, the point x = 0 is an ordinary point so that two independent series solutions are possible. For some others, the point x = 0 is a regular singular point. We do not point out which differential equation belongs to which category, because it is easy to make that decision from the look of any equation.

In this chapter, we describe the simplest possible system, viz., a free particle. To keep the notation simple, we will often pretend in this chapter that the particle moves only in one spatial dimension, so that there is only one coordinate x and its corresponding momentum p. However, the notation can be trivially modified to make the discussion applicable to motion of particles in 3D space.

Solution of free Schrödinger equation

We have introduced the Schrödinger equation in Chapter 3, which says that the time evolution of the state vector is governed by the Hamiltonian of the system:

If the Hamiltonian does not have any explicit time dependence, the solution of this equation is simply

as given in Eq. (4.15, p. 85). This solution is valid irrespective of whether the particle is free. For a free particle, the extra simplification comes from the fact that the Hamiltonian involves only the momentum operator and not the position operator. Therefore, H commutes with all components of the momentum, and so the momentum eigenstates are also energy eigenstates. This means that we can use the eigenstates of the momentum operator as the energy eigenstates and write

where Ep is the energy eigenvalue that depends on the momentum eigenvalue p. This feature can also be understood as a consequence of translational invariance.

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.