Chapter 9 presents an example of how the results of the first eight chapters can be applied to a physical theory – optics. It is all in the nature of applications, and can be omitted without any effect on the understanding of what follows. Theories of optics In the history of physics it is often the case that, when an older theory is superseded by a newer one, the older theory retains its validity, either as an approximation to the newer theory, an approximation that is valid for an interesting range of circumstances, or as a special case of the newer theory. Thus Newtonian mechanics can be regarded as an approximation to relativistic mechanics, valid when the velocities that arise are very small in comparison to the velocity of light. Similarly, Newtonian mechanics can be regarded as an approximation to quantum mechanics, valid when the bodies in question are sufficiently large. Kepler's laws of planetary motion are a special case of Newton's laws, valid for the inverse square law of force between two bodies. Kepler's laws can also be regarded as an approximation to the laws of motion derived from Newtonian mechanics when we ignore the effects of the planets on each other's motion.
Chapter 9 presents an example of how the results of the first eight chapters can be applied to a physical theory – optics. It is all in the nature of applications, and can be omitted without any effect on the understanding of what follows. Theories of optics In the history of physics it is often the case that, when an older theory is superseded by a newer one, the older theory retains its validity, either as an approximation to the newer theory, an approximation that is valid for an interesting range of circumstances, or as a special case of the newer theory. Thus Newtonian mechanics can be regarded as an approximation to relativistic mechanics, valid when the velocities that arise are very small in comparison to the velocity of light. Similarly, Newtonian mechanics can be regarded as an approximation to quantum mechanics, valid when the bodies in question are sufficiently large. Kepler's laws of planetary motion are a special case of Newton's laws, valid for the inverse square law of force between two bodies. Kepler's laws can also be regarded as an approximation to the laws of motion derived from Newtonian mechanics when we ignore the effects of the planets on each other's motion.
Historically, the scientific exploration of new phenomena has often been guided by systematic studies of observations, i.e., experimentally verifiable facts, which can be used as the basis to construct the underlying physical laws. As the apex of the investigations, one tries to identify the minimal set of fundamental assumptions – referred to as the axioms – needed to describe correctly the experimental observations. Even though the axioms form the basis to predict the system's behavior completely, they themselves have no rigorous derivation or interpretation. Thus, the axioms must be viewed as the elementary postulates that allow us to formulate a systematic description of the studied system based on well-defined logical reasoning. Even though it might seem unsatisfactory that axioms cannot be “derived,” one has to acknowledge the paramount power of well-postulated axioms to predict even the most exotic effects. As is well known, the theory of classical mechanics can be constructed using only the three Newtonian axioms. On this basis, an infinite variety of phenomena can be explained, ranging from the cyclic planetary motion all the way to the classical chaos. In this book, we are mainly interested in understanding how the axioms of classical and quantum mechanics can be applied to obtain a systematic description for the phenomena of interest. Especially, we want to understand how many-particle systems can be modeled, how quantum features of light emerge, and how these two aspects can be combined and utilized to explore new intriguing phenomena in semiconductor quantum optics.
The mathematician plays a game in which he himself invents the rules, while the physicist plays a game in which the rules are provided by nature, but as time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen. (Paul A. M. Dirac) As described in Chapter 5, classical ( Newtonian ) mechanics is centered around Hamilton's principle and the associated Newton's second law, particularly as applied to statics and dynamics of nondeformable (discrete) and deformable (continuous) bodies. Such systems are typically of O (1) size operating at O (1) speeds. Modern physics generally refers to branches of physics that have arisen since Einstein's relativity theory in the early part of the twentieth century, including special relativity, general relativity, and quantum mechanics, for example. Modern physics addresses physical phenomena that occur at very small sizes, such as atoms, or very large sizes, such as galaxies, and/or very high speeds close to the speed of light. We focus on the theory of relativity and quantum mechanics, both of which reduce to the classical mechanics limit that has occupied our considerations thus far. Relativity reduces to classical mechanics when υ ≪ c , where c is the speed of light in a vacuum, and υ is a characteristic velocity of the system; quantum mechanics reduces to classical mechanics when length scales dominant within the system are much larger than Planck's universal constant ħ.
Science has developed through a variety of investigations more or less over the time scale of human existence. On this scale, quantum mechanics is an extremely young field, existing essentially for only just over a century. Even our understanding of classical mechanics has existed for a comparatively long period—roughly having been formalized with Newton's equations published in his Principia Mathematica , in April 1686 (1687). Up until the very end of the nineteenth century, mechanics in the form of Newton's laws were comfortable, being understood and accepted by all who dealt with the motion of reasonable bodies. With the turn of the new century, however, mechanics would be shaken to its roots. The change began already on December 14, 1900, when Planck (1900) presented his new theory of black-body radiation to the German Physical Society, and this change would affect mechanics in physically small structures (and lead to quantum mechanics). Mechanics of large bodies would similarly be changed in 1905 with the arrival of Albert Einstein's (1905a) relativity theory. In this book, we will deal with only the first of these two great breakthroughs.
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.
This pedagogical monograph describes some of the fundamental views of the laws of physics. The derivations are, however, obtained from a rather unconventional point of view. The Lorentz transformations and the special theory of relativity are derived without mentioning the phenomenon of light, and the de Broglie relations in the wave-corpuscle parallelism are derived without the help of Planck's constant. By the use of Schrodinger's idea of quantization as an eigenvalue problem, the foundations of wave mechanics are discussed as a mathematical problem without reference to Planck's constant. Finally, the Kepler problem in the special theory of relativity is studied starting from the energy law, and the applications to the Hulse-Taylor binary pulsar indicate that more data about the unseen companion are needed before the interpretation of the present data may be taken as the ultimate proof of the validity of the general theory of relativity.
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.