The double-slit experiment revisited

In this chapter we turn to recent advances in our understanding of the fundamental forces of Nature. As we have said in earlier chapters, the combination of classical electromagnetism, quantum mechanics and relativity provides an astonishingly successful description of electromagnetic forces. The resulting theory is called Quantum ElectroDynamics, or QED for short. For over 50 years physicists searched for similarly successful theories to describe not only the weak forces responsible for natural radioactivity but also the strong forces that hold the nucleus together. It was not until the mid 1970s that real progress was made and these remarkable developments are the subject of this chapter.

Particle physicists now have a unified theory that combines both the electromagnetic and weak forces. The major predictions of this theory have been spectacularly verified by experiments at the CERN high-energy particle physics laboratory in Geneva. We shall describe the theory and these experiments in more detail in this chapter. But particle physicists also believe that they have at last discovered the correct theory of the strong nuclear force.

Richard Feynman and nanotechnology

In 1959, in an after dinner speech at a meeting of the American Physical Society in Pasadena, Richard Feynman set out a remarkable vision of the future in a talk entitled ‘There's plenty of room at the bottom.’ The talk was subtitled ‘an invitation to enter a new field of physics’ and marked the beginning of what is now known as ‘nanotechnology’. Nanotechnology is concerned with the manipulation of matter at the scale of a nanometre -a thousand millionth of a metre. Atoms are typically a few tenths of a nanometre in size. Feynman emphasized that such an endeavour does not need new physics:

In his talk Feynman offered two prizes of $1000 each – one ‘to the first guy who makes an operating electric motor which is only 1/64 inch cube’, and a second ‘to the first guy who can take the information on the page of a book and put it on an area 1/25000 smaller’ He had to pay out on both prizes – the first less than a year later, to Bill McLellan, a Caltech alumnus.

Barrier penetration

One of the most startling consequences of de Broglie's wave hypothesis and Schrödinger's equation was the discovery that quantum objects can ‘tunnel’ through potential energy barriers that classical particles are forbidden to penetrate. To gain some idea of what we mean by an energy barrier, let us go back to our roller coaster and look at a larger section of track, as shown in Fig. 5.1. If we start the carriage from rest, high up on the left, at A, and ignore any small frictional energy losses, we know from the conservation of energy that we shall arrive on the other side at the same height we started from, at C. As we went over the little hill B, at the bottom of the valley, the car slowed down as some of our kinetic energy was changed to potential energy in climbing the hill, but because we started much higher up, we had plenty of energy to spare to get us over the top. However, if we started the carriage from rest at A, we do not have enough energy to climb over the hill D and get to E. This is an example of an ‘energy barrier’, and we can say that the region from C to E is ‘classically forbidden’.

What is remarkable about quantum ‘particles’ is that they do not behave like these classical objects.

Science and experiment

Science is a special kind of explanation of the things we see around us. It starts with a problem and curiosity. Something strikes the scientist as odd. It doesn't fit in with the usual explanation. Maybe harder thinking or more careful observation will resolve the problem. If it remains a puzzle, it stimulates the scientist's imagination. Perhaps a completely new way of looking at things is needed? Scientists are perpetually trying to find better explanations – better in the sense that any new explanation must not only explain the new puzzle, but also be consistent with all of the previous explanations that still work well. The hallmark of any scientific explanation or ‘theory’ is that it must be able to make successful predictions. In other words, any decent theory must be able to say what will happen in any given set of circumstances. Thus, any new theory will only become generally accepted by the scientific community if it is able not only to explain the observations that scientists have already made, but also to foretell the results of new, as yet unperformed, experiments. This rigorous testing of new scientific ideas is the key feature that distinguishes science from other fields of intellectual endeavour -such as history or even economics – or from a pseudoscience such as astrology.

Rutherford's nuclear atom

Before quantum mechanics came along, classical physics was unable to account for either the size or the stability of atoms. Experiments initiated in 1911 by the famous New Zealand physicist, Ernest Rutherford, had shown that nearly all the mass and all of the positive charge of an atom are concentrated in a tiny central core that Rutherford called the ‘nucleus’. Most of the atom is empty space! A table of the relative sizes of atoms, nuclei and other quantum and classical objects is given in appendix 1. Rutherford had already won a Nobel Prize earlier, in 1908, for his work on radioactivity. Radioactivity is now known to be due to the ‘decay’ of a nucleus of certain unstable chemical elements: some radiation is given off – in the form of alpha, beta or gamma rays – and a nucleus of a different element is left behind (see Fig. 4.1). As you can imagine, it took physicists some time to disentangle what was going on, and it was Rutherford who showed that the positively charged, heavy, penetrating alpha rays were, in fact, helium atoms which had lost two electrons. Beta rays, on the other hand, were identified as electrons, and gamma rays as high energy photons. At that time, any work involving the different chemical elements was regarded as the province of chemists, and Rutherford was somewhat put out at winning the chemistry Nobel Prize.

Chapters 5–7 have shown us why and how states play such an essential role in quantum theory. States are distinguished through their labeling, which consists of a set of quantum numbers specific to the state. In general, each degree of freedom gives rise to a quantum number, where, in the present context, the phrase “degrees of freedom” refers to the number of spatial coordinates characterizing a given quantum system. The 1-D systems employed in the two chapters illustrating the postulates were thus single-degree-of-freedom systems: all the eigenstates encountered there were labeled by a single quantum number (n for bound states and k for continuum states) related to the energy of the system.

Having studied systems with a single degree of freedom, it might seem that our next step would be to consider systems with more than one degree of freedom, for example, two 1-D particles or a particle in 3-D subject to a potential, etc. Among the goals of such studies would be the assigning of labels to the eigenstates. If the assignment of quantum numbers were based solely on spatial degrees of freedom, then the preceding endeavor would indeed be a way to proceed with our development of quantum theory. However, spatial degrees of freedom are not the only source of the labels which specify eigenstates. An additional source is the sets of eigenvalues of the operators that commute with Ĥ and with each other. Such operators are the main subject of this chapter.

The quantities in physics that can be measured are conventionally denoted observables: entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems.

Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield. Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield.

The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.