By the end of the nineteenth century all the important elements of classical physics were fully mature. Newton's Laws had been applied with success to everything from industrial machines to planetary orbits. Classical thermodynamics had extended energy conservation arguments into thermal processes, and scientists including J. Willard Gibbs and Ludwig Boltzmann had developed the statistical interpretation of entropy. James Clerk Maxwell had explained all electromagnetic phenomena, from AC generators/motors to light, with the four Maxwell's equations and three material-dependent constitutive relations. There was serious discussion that physics was on the verge of being “finished” – becoming a closed system of knowledge like classical geometry with only essentially routine tasks left of determining how the known laws could be applied to engineering problems.

Yet by 1930 the foundations of mechanics, electromagnetics, and atomic-scale systems had been shaken to the core by the new revolutions in the Theory of Relativity, the photon picture of light, and quantum physics. We are still reaping the benefits of the application of these new scientific paradigms in our technology today, including semiconductor electronic devices, lasers, and GPS systems. But the description of reality that these theories provide is still deeply anti-intuitive and disturbing.

Nevertheless, quantum physics is one of the most extensively experimentally verified theories that we know. Even when quantum theory and common sense seem to diverge, quantum theory gives the right answers when tested by experiment. In the next chapter we will develop the framework of quantum theory that has been so successful when people “shut up and calculate,” as one scientist suggested as the way through the philosophical thicket of quantum physics. But in this chapter, we will spend a brief time considering the historical development and continuing paradoxical implications of quantum theory.

The photoelectric effect and photons

In 1905, Albert Einstein, who was a key figure in the revolution of modern physics, was working as a clerk in the patent office in Zurich, Switzerland. Einstein had finished his studies in physics, but had not obtained a coveted position as a university professor. He was intensely interested in the latest developments in classical physics and was aware of a few of the remaining loose ends that were resisting explanation in the classical model.

We spend most of our life in a world of macroscopic objects moving and interacting, yet many – perhaps most – people can't describe correctly the trajectory of an object thrown from a moving car. We are aware there are rules that govern motion and interactions, but what they are, when they apply, and how they can be applied to predict the motion of a pool cue ball after it collides with an object ball, for example, are a mystery to most people, judging by how often casual pool players “scratch” by pocketing the cue ball. Even engineers, who have all had at least a year-long course in physics, are often uneasy when they are asked what exactly Newton's Laws mean instead of just reciting the words. As a matter of fact, the vast majority of people are still Aristotelian thinkers in their understanding of the laws that govern motion. In this chapter we will revisit these fundamental concepts with the goal of increasing the level of comfort with the concepts of forces and motion. Newton's Laws

The traditional formulation of Newton's Laws is something like these below:

These statements are a combination of definitions and empirical (“determined by observation”) laws. For example, a force (F) is often described as a “push or pull that, if acting alone on a body, causes a constant acceleration.” This is a circular definition that makes the Second Law a meaningless statement. The First Law also has conditions on it: you can't be in an accelerating or rotating system.

This book is based on notes prepared for a graduate course “Scientific Foundations of Engineering” in the Gordon Engineering Leadership Program at Northeastern University. The elevator speech on why such a course is needed goes as follows: (1) most engineering students take all of their basic science courses during freshman year, (2) they don't like those freshman courses very much, and (3) they forget the material as quickly as they can and concentrate on the specifics of electrical engineering, or mechanical engineering, or other engineering discipline where their interests and enthusiasm lies.

This summary may be unfair to some engineering students, but most engineering students (and their professors) at least grudgingly admit that it isn't terribly far off. And, in general, this approach serves the students well through their undergraduate education process and in their industrial careers – as long as they remain specialized in their specific engineering discipline. However, consider the case where an electrical engineer is leading a multidisciplinary project. One day a mechanical engineer who reports to her walks into her office and says, “Boss, this isn't going to work – we can't get the heat out!” A conventionally trained electrical engineer isn't likely to be able to frame a single substantive question about the problem. She hasn't studied heat transport or thermodynamics since the freshman year (if at all!), and likely has forgotten anything she ever knew about the subject. The ability to frame questions which put fundamental boundaries on the problem, “What is the power load? How hot will the device get? How much blackbody emission is there at that temperature? What is the thermal conductivity of the substrate?”, will not only enable an engineering leader to quickly frame the gravity of the problem, but will undoubtedly earn her a reputation as someone with whom you want to have done your homework carefully before making rash statements about engineering limits!

Science curricula often employ a “spiral curriculum” model. In physics for example, mechanics, thermal physics, and electromagnetics are surveyed in freshman year, revisited in specialized courses in sophomore or junior year, and often studied again from a quantum statistical viewpoint in a senior class. Engineering education more often selects a “breadth” coverage of the vast range of engineering applications, rather than the depth of understanding that the spiral curriculum seeks to impart.

Once fundamental scientific laws are understood, engineering technology depends on advances in materials – materials science – more than any other factor. Historically, ages have been named for the most advanced materials technology available: Stone Age, Bronze Age, Iron Age. If you look around the room you are in, you will recognize that our current age is the age of manufactured materials: plastics, ceramics, acrylics, polyester fibers, etc. Underlying the “Information Age” are the advances in electronic materials that have made possible the computers, displays, and communications equipment that enable information transfer and processing. Selecting materials with desirable properties and then processing to enhance these properties is now standard practice. The next generation of materials will include nanoscale manipulation and fabrication of materials unlike anything found in nature. These new nanomaterials can be predicted to have properties exceeding any currently available materials.

Materials are characterized by response functions that describe the materials’ response to applied forces or fields. For example, the electromagnetic properties of materials can be characterized by the electric permittivity (electric polarization of materials in response to an applied electric field), electrical conductivity (electrical current in response to an applied electric field), and magnetic permeability (magnetic polarization in response to an applied magnetic field). One of the fundamental mechanical properties is the elasticity, which describes the deformation of the material in response to applied forces. The elasticity, as with other response functions, depends on the physics and chemistry at the atomic scale and on the structure of the material at a micro- and mesoscale, which is larger than the atoms but smaller than the macroscopic sample. For example, crystal structure, grain boundaries, and dislocations are key determining factors for elastic, plastic, and fracture properties. There are three elastic regimes: tensile/compressive elasticity, shear elasticity, and bulk elasticity. These are all special cases of a generalized elastic tensor, but we will consider them separately to begin.

Which has more internal energy: a pot of boiling water or the air in the room? This question is a good way to begin to talk about thermal phenomena. First, it introduces the concept that heat is a form of energy – the equivalence of energy and heat is the foundation of the First Law of Thermodynamics. Second, the answer depends on understanding the difference between average kinetic energy of molecules – which is what we measure by temperature – and the total internal energy of a system of molecules – which is changed by adding or removing heat. The water molecules in the pot have a higher average energy, but there may be fewer molecules in the pot than in the room air. We will be developing quantitative techniques to address this problem in this chapter, but first we need to consider the concepts of thermal equilibrium and heat capacity.

Thermal equilibrium

The first scientists who developed thermometers in the sixteenth and seventeenth centuries were surprised to find that different objects in their labs were all at the same temperature. When we touch objects that feel cold, what we are sensing is that they conduct our body heat away faster. Thus metal objects feel cooler than wood or cloth objects, but, in fact, to the first approximation everything in the room is at the same temperature, “room temperature.” This is because the room is an example of a system that is in – or near – thermal equilibrium.

The temperature is a measure of the average kinetic energy of the molecules in the system. If an object at a higher temperature comes in contact with a lower temperature object, energy will flow from the higher temperature object to the lower temperature object. This process goes on until the two objects are at the same temperature. All the objects in the room are in contact with the air and, through the air, in indirect contact with all the other objects in the room. The definition of thermal equilibrium is a system in which all the component objects are at the same temperature.

As a young professor of electrical engineering, the first author returned for a visit to the institution where he had received his PhD in Physics. His former thesis advisor, a professor of physics, was on his way to be part of a PhD oral qualifying exam committee for students in their second year of graduate study in physics. “What should I ask this guy [the qualifying exam student]?” he asked. “Why don't you ask him how a p-n junction diode works?” your first author, who had just taught a course in physical electronics, suggested. His former advisor stopped in mid-stride and said “What, do you want him to faint dead away?”

It is our suspicion that not only physics graduate students but most engineers (including electrical engineers!) would have the same difficulty in describing the operation of a p-n junction, the most fundamental and ubiquitous device in all semiconductor electronics. So how does a p-n junction, the basis of virtually every electronic device that we use every day, work? And what does that have to do with the red-hot color of a burner on an electric stove or the radiation from the Sun? These are some of the questions we will address in this chapter where we look at statistical applications of quantum theory.

As we have seen in the last chapter, many of the fundamental results of thermal phenomena can be derived by physical arguments applied to the random behavior of molecules in a system in thermal equilibrium at a temperature T. The application of statistical methods to microscopic models of a system to derive thermal phenomena is known as statistical mechanics. In this chapter we will consider the quantum modifications of statistical mechanics and the implications for thermal effects.

To review, three of the fundamental results of quantum theory are listed below.

The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures.

Postulates of quantum mechanics

QM Postulate 1: A quantum particle is described by a wave function Ψ(x, y, z, t) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ(x, y, z, t); quantities which cannot be found from the wave function are not physically meaningful.

Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.