So far, we have introduced quantum mechanics through the example of the Schrödinger equation and the spatial and temporal wavefunctions that are solutions to it. This has allowed us to solve some simple but important problems and to introduce many quantum mechanical concepts by example. Quantum mechanics does, however, go considerably beyond the Schrödinger equation. For example, photons are not described by the kind of Schrödinger equation we have considered so far, though they are undoubtedly very much quantum mechanical.

To prepare for other aspects of quantum mechanics and to make the subject easier to deal with in more complex problems, we need to introduce a more general and extended mathematical formalism. This formalism is actually mostly linear algebra. Readers probably have encountered many of the basic concepts already in subjects such as matrix algebra, Fourier transforms, solutions of differential equations, possibly (though less likely) integral equations, or analysis of linear systems, in general. For this book, we assume that the reader is familiar with, at least, the matrix version of linear algebra – the other examples are not necessary prerequisites. The fact that the formalism is based on linear algebra is because of the basic observation that quantum mechanics is apparently absolutely linear in certain specific ways as we discussed previously.

Thus far, we have dealt with the state of the quantum mechanical system as the wavefunction Ψ(r, t) of a single particle.

Quantum mechanics and real life

Quantum mechanics, we might think, is a strange subject, one that does not matter for daily life. Only a few people, therefore, should need to worry about its difficult details. These few, we might imagine, run about in the small dark corners of science, at the edge of human knowledge. In this unusual group, we would expect to find only physicists making ever larger machines to look at ever smaller objects, chemists examining the last details of tiny atoms and molecules, and perhaps a few philosophers absently looking out the window as they wonder about free will. Surely, quantum mechanics therefore should not matter for our everyday experience. It could not be important for designing and making real things that make real money and change real lives. Of course, we would be wrong.

Quantum mechanics is everywhere. We do not have to look far to find it. We only have to open our eyes. Look at some object, say a flowerpot or a tennis ball. Why is the flowerpot a soothing terra-cotta orange color and the tennis ball a glaring fluorescent yellow? We could say each object contains some appropriately colored pigment or dye, based on a material with an intrinsic color, but we are not much further forward in understanding. (Our color technology would also be stuck in medieval times, when artists had to find all their pigments in the colors in natural objects, sometimes at great cost.)

In this section, we return to the harmonic oscillator and consider it in a mathematically more elegant way. This approach leads to the introduction of “raising” and “lowering” operators that take us from one harmonic oscillator state to another. The introduction of these operators allows us to rewrite the harmonic oscillator mathematics quite economically. We then show that the electromagnetic field for a given mode can also be described in a manner exactly analogous to a harmonic oscillator. In this case, we describe the states of this generalized harmonic oscillator in terms of the number of photons per mode, with that number corresponding exactly to the quantum number for the corresponding harmonic oscillator state. The raising and lowering operators are now physically interpreted as “creation” and “annihilation” operators for photons and are key operators for describing electromagnetic fields.

By this process, we can describe the electromagnetic field quantum mechanically, rather than our previous semiclassical use of quantum mechanical electron behavior but with classical electric and magnetic fields. We say we have “quantized” the electromagnetic field. This quantization is then the basis for all of quantum optics. This approach also prepares us for discussions in subsequent chapters of fermion operators and of the full description of stimulated and spontaneous emission.

For teachers

The entire material in this book could be taught in a one-year course. More likely, depending on the interests and goals of the teacher and students and the length of time available, only some of the more advanced topics will be covered in detail. In a two-quarter course sequence for senior undergraduates and for engineering graduate students at Stanford, the majority of the material here will be covered, with a few topics omitted and some covered in lesser depth.

The core material (Chapters 1–5) on Schrödinger's equation and on the mathematics behind quantum mechanics should be taught in any course. Chapter 4 gives a more explicit introduction to the ideas of linear operators than is found in most texts. Chapter 4 also explains and introduces Dirac notation, which is used from that point onward in the book. This introduction of Dirac notation is earlier than in many older texts, but it saves considerable time thereafter in describing quantum mechanics. Experience teaching engineering students in particular, most of whom are quite familiar with linear algebra and matrices from other applications in engineering, shows that they have no difficulties with this concept.

Aside from that core, there are many possible choices about the sequence of material and about what material needs to be included in a course. The prerequisites for each chapter are clearly stated at the beginning of the chapter.

The hydrogen atom is a very important problem in quantum mechanics. It is also a problem described by Schrödinger's equation that is mathematically exactly solvable (if we neglect a few small corrections that lie beyond the simple Schrödinger equation description). These solutions give the basis for our conceptual understanding of atoms and molecules, as well as much of the notation. In engineering, the hydrogen atom solutions also can be used as a model to explain a phenomenon (i.e., Wannier excitonic effects) that is very important in the optical absorption in most semiconductor optoelectronic devices.

The hydrogen atom problem is additionally an excellent tutorial one to take us beyond the simple one-dimensional spatial problems we have used as examples so far. In particular, it is a concrete example of the angular momentum behavior we discussed in the previous chapter. Also, unlike all the problems dealt with so far, the hydrogen atom involves two particles (the electron and the proton), not just one. The hydrogen atom, therefore, gives us an introduction to how we can handle more than one particle (though, unfortunately, problems containing more than two particles are almost always not exactly solvable). Additionally, we use this problem to illustrate another technique that is often used in deriving the solutions for those Schrödinger problems that can be solved exactly: namely, series solution of differential equations.

One aspect of quantum mechanics that is very different from the classical world is that particles can be absolutely identical – so identical that it is meaningless to say which is which. This “identicality” has substantial consequences for what states are allowed, quantum mechanically, and in the counting of possible states. Here, we examine this identicality, introducing the concepts of fermions and bosons and the Pauli exclusion principle that lies behind so much of the physics of materials.

Scattering of identical particles

Suppose we have two electrons in the same spin state, electrons that, for the moment, we imagine we can label as electron 1 and electron 2. We write the spatial coordinates of electron 1 as r1 and those of electron 2 as r2. As far as we know, there is absolutely no difference between one electron and another. They are absolutely interchangeable. We might think, because of something we know about the history of these electrons, that it is more likely that we are looking at electron 1 rather than electron 2, but there is no way by making a measurement so that we can actually know for sure at which one we are looking.

We could imagine that the two electrons were traveling through space, each in some kind of wavepacket. The wavepackets might each be quite localized in space at any given time.

Thus far, we have worked with a quantum mechanical wave for electrons and similar particles, we have worked with classical waves for electric and magnetic fields, and we have introduced a quantum mechanical way of looking at electric and magnetic fields through the use of boson annihilation and creation operators. The use of these operators for boson modes led to the quantization of the electromagnetic field into photons. These operators naturally behaved in such a way as to give the photons the properties we expect of bosons, allowing any zero or positive integer number of photons in a mode. We will find that we can also introduce annihilation and creation operators for fermions and it is the purpose of this chapter to make this introduction. These operators lead to the natural quantization of the number of fermions possible in a fermion “mode” or single-particle state, limiting us to zero or one as required. Analogously to the boson operator description of the electromagnetic wave with field operators, we can also describe the quantum mechanical wave associated with electrons and similar particles in terms of the fermion operators.

Just as the quantization of the electromagnetic wave led to the appearance in our quantum mechanics of boson particles, we can view this introduction of the fermion annihilation and creation operators as leading to the appearance in our quantum mechanics of fermion particles.

If the world of quantum mechanics is so different from everything we have been taught before, how can we even begin to understand it? Miniscule electrons seem so remote from what we see in the world around us that we do not know what concepts from our everyday experience we could use to get started. There is, however, one lever from our existing intellectual toolkit that we can use to pry open this apparently impenetrable subject, and that lever is the idea of waves. If we just allow ourselves to suppose that electrons might be describable as waves and follow the consequences of that radical idea, the subject can open up before us. Astonishingly, we will find we can then understand a large fraction of those aspects of our everyday experience that can only be explained by quantum mechanics, such as color and the properties of materials. We will also be able to engineer novel phenomena and devices for quite practical applications.

On the face of it, proposing that we describe particles as waves is a strange intellectual leap in the dark. There is apparently nothing in our everyday view of the world to suggest we should do so. Nevertheless, it was exactly such a proposal historically (i.e., de Broglie's hypothesis) that opened up much of quantum mechanics. That proposal was made before there was direct experimental evidence of wave behavior of electrons.

Up to this point, we have essentially presumed that the state of a system, such as an electron or even a hydrogen atom, can be specified by considering a function in space and time, a function we have called the wavefunction. That description in terms of one quantity, the scalar amplitude of the wavefunction, turns out not to be sufficient to describe quantum mechanical particles. We find that we also need to specify amplitudes for the spin of the particle.

The idea that we would need additional degrees of freedom to describe a system completely is not itself unusual. For example, in classical mechanics, we might use position as a function of time to describe an object such as a football, but that would not be a complete description; we might need to add a description of the rotation of the football so that we could calculate the expected future dynamics of the ball. The rotation of the football is still something we can describe purely in terms of the position of points on the football and we might view it as being loosely analogous to the angular momentum of a quantum mechanical particle such as a hydrogen atom. We might also, however, need to add the color of the ball to its description if we were showing the ball on television and the manufacturer of the ball would care very much that the correct name was clearly displayed on the ball.

The essential aspects of the quantum mechanics introduced in the first half of this book mostly come from the conceptual development of quantum mechanics from ∼1900 to the early 1930s. The following is a very brief (and substantially incomplete) chronology of some of the most cited early development. Of course, the whole history is much richer than this and is worthy of a much deeper treatment.

1900 Max Planck postulates that the energy in light comes in quanta of size hν, where h is Planck's constant. This solves a famous problem of classical physics, the “ultraviolet catastrophe,” in which otherwise the thermal distribution of energy in light should keep on increasing without limit at ever shorter wavelengths.

1905 Albert Einstein postulates the photon to explain the photoelectric effect. This now clearly introduces the concept of wave-particle duality for light.

1913 Neils Bohr proposes the quantization of angular momentum, which gives the Bohr model of the hydrogen atom that, especially when further developed by Sommerfeld in 1916 and by Debye, successfully explains major features of the hydrogen atom's energy levels and spectra, though it leaves other aspects unexplained, especially why it is that the orbiting electrons of this model are not continuously emitting radiation.

1922 Otto Stern and Walther Gerlach show in their experiment the quantized nature of electron spin and additionally expose the key difficulty of the measurement problem in quantum mechanics.

Many interesting quantum mechanical problems can be reduced to one-dimensional mathematical problems. This reduction is often possible because the problem, though truly three-dimensional, can be mathematically separated. For example, the three-dimensional hydrogen atom also mathematically separates to leave a radial equation that looks like a onedimensional effective Schrödinger equation. Most problems associated with electrons and planar surfaces or layered structures can be handled with one-dimensional models. Examples include field emission of electrons from planar metallic surfaces and most problems associated with semiconductor quantum well structures.

One-dimensional problems can be solved by a number of techniques. Here, we discuss one of these, the transfer matrix technique, and we also derive one key result of the so-called WKB method. We concentrate on the use of such techniques for solving tunneling problems, so we start with a brief discussion of tunneling rates.

Tunneling probabilities

Suppose we have a barrier, shown in Fig. 11.1 as a simple rectangular barrier. Electrons are incident on the barrier from the left. Some are reflected and some are transmitted. We presume that the electron energy E is less than the barrier height V0 so that we are discussing a tunneling problem. We already know how to solve this problem quantum mechanically for the simple case of a rectangular barrier, with notation as shown in the figure. We discuss now how to solve such problems for more complex forms of barrier.