This chapter introduces two approaches to quantum mechanics, one is to follow the historical development of the theory from the first indications that the whole fabric of classical mechanics and electrodynamics should be held in doubt to the resolution of the problem. The other approach is to stand back at a point late in the development of the theory and see its underlying theoretical structure. The chapter shows that in the first approach the theory is seen gradually emerging from confusion and dilemma, while the second is a more formal approach that can still be exciting and compelling in a different sense. The chapter highlights the logic and elegance in a scheme that starts from only a few postulates, yet reveals a rich, experimentally verifiable structure. It traces some of the most fundamental concepts of the nature of matter and its behaviour that were overthrown and replaced by a puzzling but powerful new description.
This chapter introduces two approaches to quantum mechanics, one is to follow the historical development of the theory from the first indications that the whole fabric of classical mechanics and electrodynamics should be held in doubt to the resolution of the problem. The other approach is to stand back at a point late in the development of the theory and see its underlying theoretical structure. The chapter shows that in the first approach the theory is seen gradually emerging from confusion and dilemma, while the second is a more formal approach that can still be exciting and compelling in a different sense. The chapter highlights the logic and elegance in a scheme that starts from only a few postulates, yet reveals a rich, experimentally verifiable structure. It traces some of the most fundamental concepts of the nature of matter and its behaviour that were overthrown and replaced by a puzzling but powerful new description.
Quantum mechanics is essentially a 20th century development that is based on a number of observations that defied classical explanations. While some of these experiments have semiclassical explanations, the triumph of quantum mechanics is that it gives precise verification for an overwhelming number of experimental observations. Its extensions into relativistic quantum mechanics through quantum electrodynamics (QED), electro-weak theory, and quantum chromodynamics (QCD) have led us to the Standard Model of today. While the number of unanswered questions remains approximately constant at each stage of development, the number of answered questions that relate theory with experiment continues to grow rapidly. In this chapter, some of the historical landmarks in the development of the theory are noted. The resolution of the dilemmas presented by classical theory will be dealt with in later chapters, but these are listed to motivate the break from classical mechanics. Because of the apparently unphysical nature of the postulates upon which quantum mechanics is founded, we supply motivation and justification for this break. In the first part of this chapter, we will review some of the experiments that confounded classical theory, and then introduce a formalism that provides some rationale for the postulates upon which quantum mechanics is based. In the end, these postulates will stand on their own.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
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.
Quantum theory has passed through three distinct stages during its evolution. The first is generally referred to as ‘old quantum theory’. During a period roughly between 1900 and 1925 Planck, Bohr and Einstein, among others, grappled with various experimental and theoretical crises by patching up classical mechanics with some new quantum ideas applied as band-aids. This is not to disparage the work; it was vitally important, and you can only do what is possible at the time. The quantum world is very different to the picture painted by classical physics. That being the case and given the extreme nature of some of these differences, we might question the wisdom of accepting the quantum view. Scientists can be quite disparaging about philosophy, sometimes while in the process of espousing, quite stridently, a philosophical position of their own. The chapter also presents an overview on the key concepts discussed in this book.