# I THE FIRST QUANTUM AGE

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## Quantum theory—basic ideas

There is no doubt that quantum theory is a theory of tremendous power and importance; indeed it is probably the most successful physical theory of all time! It tells us a vast amount about the properties of atoms, molecules, nuclei, and solids, and it is also essential for any understanding of astrophysics. In fact we can go further: quantum theory is not only central in virtually every field of physics, but it is also the theoretical background to the whole of modern chemistry, and is becoming of increasing importance in our quest to understand biology at the molecular level. So we can sum up by saying that it is supremely important for any kind of understanding of how the Universe works.

In addition, though, it has generated a large number of technical applications. In particular there are very many specifically quantum devices, starting with the transistor, and moving on through all the related solid state devices that have followed it, right up to the today’s famous integrated circuits and ‘computer chips’. And we must certainly not forget the ubiquitous laser. Devices based on quantum principles have had an enormous range of applications—in industry, in medicine, in computers, in all the devices we take for granted in home, shop or factory—with the result that, in addition to its intellectual interest, quantum theory manages to contribute a substantial fraction of the economic output of the developed nations.

However, it must be admitted that quantum theory is admittedly difficult—in two rather distinct ways. The first difficulty is that it is mathematically complex; in this book, though, we shall do our best to explain the essential ideas without bringing in too many of these technical details. But over and above the rather advanced mathematics, the second difficulty is that the theory has a number of elements that are very difficult to accept or even understand, at least from the standpoint of the pre-quantum or so-called classical physics that we are all used to, or even just from a commonsense approach to ‘physical reality’.

In Part I of this book, we shall describe some of these conceptually challenging aspects of the theory, and also the way in which these challenges have been met in what we call the First Quantum Age. Our approach is not historical, but we do include biographical boxes relating the ideas described to the physicists who discovered or elucidated them. A fuller account of both the ideas and the history is given in [1].

In the rest of this chapter, we make a few introductory remarks about quantum ideas. To start with, we should make the point that nearly everything we discuss follows from the central law of non-relativistic quantum theory, the famous Schrodinger equation, though it must be supplemented by the Born probability rule. Both the Schrodinger equation and the Born rule will be discussed shortly, in Chapters 3 and 4, respectively.

However, two general points on this all-important Schrodinger equation can be mentioned right away. First, we said that it is non-relativistic. This means that the discussion is restricted to the case where the speeds of the bodies being considered are much less than that of light, always called c. which has the value 3 x 10^{8} ms^{-1}. In practice this is not really such an important problem as one might expect, since in many of the most important applications of quantum theory the particles do indeed have speeds much less than *c*. It would actually be possible to make the discussion relativistic by going beyond the Schrodinger equation, but this would only be achieved at the expense of considerable extra complication, and as it does not change the conceptual status of the theory to any extent, we shall stick to the non-relativistic case in this book.

What is actually more relevant for our account is that many of the experiments that we shall discuss involve the passage of photons from one point to another. As we shall see later in the following chapter, photons must be regarded as particles of light and therefore obviously travel at speed c. Our previous paragraph might seem to rule out consideration of photons, but in fact it turns out that we are able to discuss these experiments in a totally satisfactory way using the Schrodinger equation.

The second point is that the fundamental equation of quantum theory is known as the *time-dependent* Schrodinger equation (TDSE). Shortly we shall meet the *time-independent* Schrodinger equation (TISE). This has no independent status, being obtained directly from the TDSE for situations where the physics does not depend on time. However, the TISE is exceptionally useful in its own right, as we shall see; indeed, although it is less fundamental than the TISE, it is probably used much more.

We now come to a point that is rather subtle but is of very great importance for much of the rest of the book. This is that in quantum theory we must accept that the Schrodinger equation gives much important information about the system in question, but not all the information we are used to in classical (pre-quantum) physics.

In classical physics, for example, we take it for granted that both the position and the momentum of a particle exist and have precise values at all times, and that the central set of equations of classical mechanics, Newton’s laws, does indeed use and provide values for both these quantities simultaneously. However, in quantum theory the situation is very different. As we shall see, the Schrodinger equation can give us precise values for one or other of the position and momentum of a particle, but if it gives us a precise value for its position, it can say nothing at all about its momentum, and if it gives us a precise value for its momentum, it can say nothing at all about its position.

Of course to say that the Schrodinger equation cannot provide precise values for both position and momentum does not necessarily mean that the particle itself does not actually *possess* such values. It might indeed seem most natural to assume that it *does* possess the values, and it is merely a limitation of the Schrodinger equation that it cannot provide them.

However, a striking addition to the statement of the Schrodinger equation has often been assumed, virtually universally, in fact, in the First Quantum Age. It is an important part of what is called the Copenhagen interpretation of quantum theory, which we shall meet shortly. This addition says that if the value of a particular quantity is not provided by the Schrodinger equation, such a value actually does not exist in the physical world. This is clearly a stronger statement than saying, for example, that the value of the quantity might exist but the Schrodinger equation cannot provide it (as we suggested in the previous paragraph), or even that it exists but that we are, for some reason, prohibited from knowing it.

Thus, in our example above, it may be that, at a particular time, the Schrodinger equation gives us a value for the position of the particle. Actually we can put that more precisely—we are saying that, at this time, the so-called wave-function of the particle relates to a precise value of position. (We shall study the idea of the wave-function more formally in Chapter 3 . ) Then, as we have said, the Schrodinger equation, or more specifically the form of the wave-function at this time, can tell us absolutely nothing about the momentum of the particle. That is a central and non-controversial feature of quantum theory.

The additional element we mentioned in the previous paragraph, which is much more controversial, says that we must then assume that the momentum of the particle just does not have a value; it is *indeterminate. *Similarly if the wave-function provides us with a precise value of momentum at a particular time, then we must just take it for granted that the position of the particle just does not have a value at this particular time.

In Einstein’s terms, we are talking about *completeness.* We are saying that the Schrodinger equation is *complete* in the sense that nothing exists in the physical universe that cannot be obtained from the equation—if you can’t get it from the wave-function, it just doesn’t exist! It must be said that Einstein himself strongly disagreed with the idea that the Schrodinger equation was complete. He believed that there were elements of reality lying outside the remit of the Schrodinger equation. This disagreement of Einstein with the Copenhagen interpretation of quantum theory is extremely important and it will be investigated in depth later in the book.

As we have said, this statement of completeness clearly constitutes a very important difference between quantum theory and classical physics. In classical physics it is naturally taken for granted that all physical quantities have particular values at all times, but this is not the case in

**Albert Einstein (1879-1955)**

Fig. 1.1 **Albert Einstein in 1921 [courtesy of Hebrew University of Jerusalem]**

Einstein is most famous for his work on relativity, his special theory of relativity, which was published in 1905, and the general theory, which included gravitation and which followed in 1916, but his work on the quantum theory was equally important. Einstein’s most important contribution to quantum theory was the argument in 1905 (his *annus mirabilis or* ‘wonderful year’) that suggested the existence of the *photon;* the fact that light had a particle-like as well as a wavelike nature was the first of a series of conceptual difficulties that were to plague the quantum theory. His idea of the photon emerged from fundamental studies on the theory of radiation, but it was able to predict and explain the essential features of the photoelectric effect, and it was for this work that Einstein was awarded the Nobel Prize for physics in 1921.

Thomas Kuhn has argued [2] that Einstein’s work in the first decade of the century was of crucial importance in clarifying the full significance of quantization

and discreteness that had been demonstrated but not really explained or even understood by Planck in 1900.

Einstein made other important contributions to quantum theory with his work on the specific heat of gases and solids, and his discovery in 1917 of *stimulated emission,* which was the basis of the *laser.* Then in 1924, Einstein contributed to the development of the ideas of Satyendra Nath Bose that led to the discovery of the Bose-Einstein statistics that are obeyed by particles such as photons, and also by the recently discovered Bose-Einstein condensate, where quantum effects are observable on a large or macroscopic scale rather than a microscopic one.

During the 1920s, Einstein gave crucial support to the ideas of de Broglie and Schrodinger, but his unwillingness to accept Bohr’s *complementarity* led to his subsequent ideas being dismissed. Only recently has it gradually been realized that many of his arguments were profound and important [3]. In particular his ideas stimulated the important work of John Bell.

Einstein did not develop strong links to any country. He was born in Germany, but lived and studied in Switzerland from 1895, famously being employed in the Swiss Patent Office in Berne from 1902. Following his exceptionally important early work, he took up academic appointments in Zurich and then Prague from 1909, and then in 1914 he returned to Germany to the supremely important post of Director of the Kaiser Wilhelm Institute of Physics, Professor in the University of Berlin and a member of the Prussian Academy of Sciences. However, he always disliked German militarism, and in 1933 he was driven out by the Nazis, emigrating to the United States and taking up a position at the Institute for Advanced Study in Princeton, where he lived until his death.

quantum theory. We can put this slightly differently. Since quantum theory tells us that physical quantities do not necessarily possess values prior to measurement, we can say that measurement, at least in some sense, actually creates the value measured. Certainly the question of *measurement* is crucial in quantum theory. In contrast we can say that in classical physics it is, of course, taken for granted that physical quantities exist totally independently of whether we measure them, or even whether we *can* measure them.

To return to what we have called this additional feature of the Copenhagen interpretation, the opposing point of view is that there may be additional variables*—hidden variables* or *hidden parameters*— which cannot be obtained from the wave-function, but nevertheless can give us additional information about the physical quantities related to individual systems; in other words, the Schrodinger equation is *not *complete. Thus the orthodox view in the First Quantum Age has been that hidden variables did not exist, but the question of their possible existence became of increasing importance in the New Quantum Age.

**References**

- 1. A. Whitaker,
*Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information*(Cambridge: Cambridge University Press, 2006). - 2. T. S. Kuhn,
*Black-Body Theory and the Quantum Discontinuity 1894-1912*(Chicago: University of Chicago Press, 1987). - 3. D. Home and A. Whitaker,
*Einstein’s Struggles with Quantum Theory: A Reappraisal*(New York: Springer, 2008).

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