The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter.

According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.

Quantum mechanics has evolved from a subject of study in pure physics to one with a vast range of applications in many diverse fields. Some of its most important applications are in modern solid state electronics and optics. As such, it is now a part of the required undergraduate curriculum of more and more electrical engineering, materials science, and applied physics schools. This book is based on the lecture notes that I have developed over the years teaching introductory quantum mechanics to students at the senior/first year graduate school level whose interest is primarily in applications in solid state electronics and modern optics.

There are many excellent introductory text books on quantum mechanics for students majoring in physics or chemistry that emphasize atomic and nuclear physics for the former and molecular and chemical physics for the latter. Often, the approach is to begin from a historic perspective, recounting some of the experimental observations that could not be explained on the basis of the principles of classical mechanics and electrodynamics, followed by descriptions of various early attempts at developing a set of new principles that could explain these ‘anomalies.’ It is a good way to show the students the historical thinking that led to the discovery and formulation of the basic principles of quantum mechanics. This might have been a reasonable approach in the first half of the twentieth century when it was an interesting story to be told and people still needed to be convinced of its validity and utility.

While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter.

Definitions of the density operator and the density matrix

Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.

What is quantum mechanics and what does it do?

In very general terms, the basic problem that both classical Newtonian mechanics and quantum mechanics seek to address can be stated very simply: if the state of a dynamic system is known initially and something is done to it, how will the state of the system change with time in response?

In this chapter, we will give a brief overview of, first, how Newtonian mechanics goes about solving the problem for systems in the macroscopic world and, then, how quantum mechanics does it for systems on the atomic and subatomic scale. We will see qualitatively what the differences and similarities of the two schemes are and what the domain of applicability of each is.

Brief overview of classical mechanics

To answer the question posed above systematically, we must first give a more rigorous formulation of the problem and introduce the special language and terminology (in double quotation marks) that will be used in subsequent discussions. For the macroscopic world, common sense tells us that, to begin with, we should identify the “system” that we are dealing with in terms of a set of “static properties” that do not change with time in the context of the problem. For example, the mass of an object might be a static property. The change in the “state” of the system is characterized by a set of “dynamic variables.

With the basic quantum theory of atomic systems developed in the previous chapters, it is now possible to address the question, at least in a qualitative way, of how atoms can be held together to form molecules and crystalline solids. The explanation is based on the time-independent Schrödinger equation, which is solved on the basis of time-independent perturbation theory.

When the atoms are brought together, the electrons and ions in the atoms interact also with the positive charges in the nuclei and the electrons of the neighboring ions. Quantum mechanically, it may be energetically more favorable for the atoms to form molecular complexes than to exist as separate atoms. A simple molecular orbital theory of “covalent bonded” diatomic molecules is introduced. This model can lead to a qualitative understanding of, for example, some simple sp, sp2, or sp3 bonded organic molecules, and sp3bonded tetrahedral complexes that are the basic building blocks of such important IV–IV elemental semiconductors as Si and Ge and various III–V and II–VI compound semiconductors such GaAs, GaP, ZnS, and CdS. The basic geometry of the atomic orbitals of the constituent atoms determines the structures of the tetrahedral complexes, which in turn determine the crystalline structures of the solids. Of particular interest are semiconductors with broad applications in electronics and photonics.

Time-independent perturbation theory

The key to solving the time-dependent Schrödinger equation is to solve the corresponding time-independent Schrödinger equation. Yet, in the vast majority of cases, the time-independent Schrödinger equation cannot be solved exactly analytically.

Basic scientific theories usually start with a set of hypotheses or “postulates.” There is generally no logical reason, apart from internal consistency, that can be given to justify such postulates absolutely. They come from ‘revelations’ in the minds of ‘geniuses,’ most likely with hints from Nature based on extensive careful observations. Their general validity can only be established through experimental verification. If numerous rigorously derived logical consequences of a very small set of postulates all agree with experimental observations without exception, one is inclined to accept these postulates as correct descriptions of the laws of Nature and use them confidently to explain and predict other natural phenomena. Quantum mechanics is no exception. It is based on a few postulates. For the purpose of the present discussion, we begin with three basic postulates involving: the “state functions,” “operators,” and “equations of motion.”

In this chapter, this set of basic postulates and some of the corollaries and related definitions of terms are introduced and discussed. We will first simply state these postulates and introduce some of the related mathematical tools and concepts that are needed to arrive at their logical consequences later. To those who have not been exposed to the subject of quantum mechanics before, each of these postulates taken by itself may appear puzzling and meaningless at first. It should be borne in mind, however, that it is the collection of these postulates as a whole that forms the foundations of quantum mechanics.

Some of the most important applications of quantum mechanics are in semiconductor physics and technology based on the properties of electrons in a periodic lattice of ions. This problem is discussed on the basis of the nearly-free-electron model of the crystalline solids in this chapter. In this model, the entire solid is represented by a quantum well of macroscopic dimensions. The spatially-varying electron potential due to the periodic lattice of ions inside the well is considered a perturbation on the free-electron states leading to the Bloch states and the band structure of the semiconductor. The concepts of effective mass and group velocity of the electrons and holes in the conduction and valence bands separated by an energy-gap are introduced. The electrons and holes are distributed over the available Bloch states in these bands depending on the location of the Fermi level according to Fermi statistics. The transport properties of these charge-carriers and their influence on the electrical conductivity of the semiconductor are discussed. When impurities are present, the electrical properties can be drastically altered, resulting in n-type and p-type semiconductors. The p–n junction is a key element in modern semiconductor electronic and photonic devices.

Molecular orbital picture of the valence and conduction bands of semiconductors

Atoms can be brought together to form crystalline solids through a variety of mechanisms. Most of the commonly used semiconductors are partially covalently and partially ionically bonded crystals of diamond or zincblende structure.

An optical fiber is basically a cylindrical dielectric waveguide with a circular cross section where a high-index waveguiding core is surrounded by a low-index cladding. Optical fibers are usually made of silica (SiO2) glass. The index step and profile are controlled by the concentration and distribution of dopants. For example, the core can be doped with germania (GeO2) or alumina (Al2O3) or other oxides, such as P2O5 or TiO2, for a slightly higher index than that of a silica cladding. Alternatively, to take advantage of low-loss pure silica, the cladding can be doped with fluorine for a slightly lower index while the core contains undoped pure silica. Silica fibers are ideal for light transmission in the visible and near-infrared regions because of their low loss and low dispersion in these spectral regions. They are therefore suitable for optical communications and most laser applications in this range of the spectrum. Optical fibers made of other materials are also developed for special applications. For example, low-cost plastic fibers can be used for short-distance interconnections between personal computers and printers in offices. Fibers composed of ZrF4, BaF2, AlF3, LiF3, and other fluorides have a low loss in the range of 2–4 μm in the mid infrared. They can be used for mid-infrared optical communication or medical applications. Fibers for other spectral regions, such as the 10-μm region of CO2 laser wavelengths, are also developed.

Optical fibers have a wide range of applications. Owing to their low losses and large bandwidths, their most important applications are fiber-optic communications and interconnections. Other important applications include fiber sensors, guided optical imaging, remote monitoring, and medical applications.

A photodetector is a device that converts an optical signal into a signal of another form. Most photodetectors convert optical signals into electrical signals, in the form of either current or voltage, that can be further processed or stored. All photodetectors are squarelaw detectors that respond to the power or intensity, rather than the field amplitude, of an optical signal. Based on the difference in the conversion mechanisms, there are two classes of photodetectors: photon detectors and thermal detectors. Photon detectors are quantum detectors based on the photoelectric effect, which converts a photon into an emitted electron or an electron–hole pair; a photon detector responds to the number of photons absorbed by the detector. Thermal detectors are based on the photothermal effect, which converts optical energy into heat; a thermal detector responds to the optical energy, rather than the number of photons, absorbed by the detector. Because of the difference in their fundamental mechanisms, there are a number of important differences in the general characteristics of these two classes of detectors.

The response of a photon detector is a function of optical wavelength with a longwavelength cutoff, whereas that of a thermal detector is wavelength independent. A photon detector can be much more responsive than a thermal detector in a particular spectral region, which typically falls somewhere within the range from the near ultraviolet to the near infrared. In comparison, a thermal detector normally covers a wide spectral range from the deep ultraviolet to the far infrared with a nearly constant response. Photon detectors can be made extremely sensitive.

The functioning of electro-optic, magneto-optic, and acousto-optic devices discussed in earlier chapters is based on the fact that the optical properties of a material depend on the strength of an electric, magnetic, or acoustic field that is present in an optical medium. At a sufficiently high optical intensity, the optical properties of a material also become a function of the optical field. Such nonlinear response to the strength of the optical field results in various nonlinear optical effects. Nonlinear optics is an established broad field with applications covering a very wide range. The most important nonlinear optical devices are optical frequency converters. The frequency-converting function of such devices is uniquely nonlinear and is difficult, if not impossible, to accomplish by other means in the absence of optical nonlinearity. Other unique nonlinear optical devices include all-optical switches and modulators. Many interesting nonlinear optical phenomena, such as optical solitons, stimulated Raman scattering, and optical phase conjugation, also find useful applications.

Optical nonlinearity

The origin of optical nonlinearity is the nonlinear response of electrons in a material to an optical field as the strength of the field is increased. Macroscopically, the nonlinear optical response of a material is described by a polarization that is a nonlinear function of the optical field. In general, such nonlinear dependence on the optical field can take a variety of forms. In particular, it can be very complicated when the optical field becomes extremely strong.