Introduction

In spite of its wonderful agreement with every experiment performed so far, standard quantum theory (SQT) fails to describe events and to define the circumstances under which such events occur [171]. It is the linearity of Schrödinger evolution that lies, as we have seen, at the root of the problem. A state vector always evolves to become a linear superposition of the states corresponding to several possible outcomes of a measurement, and it is only when an experiment is actually done that one of these possible outcomes is realized at a particular instant. Thereupon the state has to be changed to the one corresponding to the particular outcome in order to follow its subsequent evolution. This additional information is not contained in the theory and has to be obtained from outside. This means that the theory is unable to predict when an event will occur. All it can predict is that if an event occurs, the possible outcomes and their probabilities are such and such. Since events do occur in every experiment, there is something missing from the theory.

Since a state vector can be written as the linear sum of a complete set of basis states and these basis states can be chosen in a number of ways, each of which corresponds to a different set of outcomes, the theory also fails to tell us how to choose the preferred basis.

Introduction

Ever since 1916/17 when Einstein argued [45] [46] that spontaneous emission must occur if matter and radiation are to achieve thermal equilibrium, physicists have believed that spontaneous emission is an inherent quantum mechanical property of atoms and that excited atoms inevitably radiate. This view, however, overlooks the fact that spontaneous emission is a consequence of the coupling of quantized energy states of atoms with the quantized radiation field, and is a manifestation of quantum noise or of emission ‘stimulated’ by ‘vacuum fluctuations’. An infinity of vacuum states is available to the photon radiated by an excited atom placed in free space, leading to the effective irreversibility of such emissions. If these vacuum states are modified, as for example by placing an excited atom between closely spaced mirrors or in a small cavity (essentially the Casimir effect), spontaneous emission can be greatly inhibited or enhanced or even made reversible. Recent advances in atomic and optical techniques have made it possible to control and manipulate spontaneous emission. A whole new branch of quantum optics called ‘cavity QED’ has developed since 1987 utilizing these dramatic changes in spontaneous emission rates in cavities to construct new kinds of microscopic masers or micromasers that operate with a single atom and a few photons or with photons emitted in pairs in a two-photon transition.

Introduction

In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons.

Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [a, b]. Three kinds of time have been defined in this context. One, called the transmission time τT(a, b), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τR(a, b) is the average time spent within the barrier region by the particles that are eventually reflected.

Introduction

The quantum theory of atoms and molecules had its origin in the famous 1913 paper of Bohr [186] in which he suggested that the interaction of radiation and atoms occurred with the transition of an atom from one stationary internal state to another, accompanied by the emission or absorption of radiation of a frequency determined by the energy difference between these states. These transitions came to be known as ‘quantum jumps’. Since all experiments until the late 1970s used to be carried out with large ensembles of atoms and molecules, these jumps were masked and could not be directly observed; they were only inferred from spectroscopic data. In fact, with the advent of quantum mechanics they eventually came to be regarded as artefacts of Bohr's simple-minded semi-classical model. But with the availability of coherent light sources and single ions prepared in ion traps [187], [188], and optically cooled [189], [190], the issue has been reopened with the experimental demonstration of quantum jumps in single ions [191], [192], [193]. These experiments have also opened up the contentious issue of wave function collapse or reduction on measurement, as they can be regarded as making collapse visible on the oscilloscope screen [191].

Evidence of the discrete nature of quantum transitions in a single quantum system had been accumulating from the observation of photon anti-bunching in single-atom resonance fluorescence [194], the tunneling of single electrons in metal–oxide-–semiconductor junctions [195] and spin-flips of individual electrons in a Penning trap [196].

Careful experiments with radiation, molecules, atoms and subatomic systems have convinced physicists over the years that the laws governing them (embodied in quantum mechanics) are quite different from those governing familiar objects of everyday experience (embodied in classical mechanics and electrodynamics). Quantum mechanics has turned out to be a very accurate and reliable theory though, even after more than seventy years of its birth, its interpretation continues to intrigue physicists and philosophers alike. Thanks to enormous technological advances over the last couple of decades, it has now become possible actually to perform some of the gedanken experiments that the pioneers had thought of to highlight the counterintuitive and bizarre consequences of quantum theory. So far, quantum mechanics has emerged unscathed in every case, and continues to defy all attempts at falsification. The spectacular success of its working rules has spurred physicists in recent times to grapple seriously with its foundational problems, leading to new theoretical and technological advances.

On the other hand, general relativity, the paradigm of classical field theory, continues to remain as accurate and reliable as quantum mechanics in its own domain of validity, namely, the large-scale universe. To wit, the agreement between the predictions of general relativity and observation of the energy loss due to gravitational waves emitted by binary pulsars is just as impressive as the agreement between the prediction of quantum electrodynamics and the measured value of the Lamb shift in atoms.

Introduction

Quantum theory predicts the striking and paradoxical result that when a system is continuously watched, it does not evolve! Although this effect was noticed much earlier by a number of people [146], [147], [148], [149], [150], it was first formally stated by Misra and Sudarshan [151] and given the appellation ‘Zeno's paradox’ because it evokes the famous paradox of Zeno denying the possibility of motion to a flying arrow. It is as startling as a pot of water on a heater that refuses to boil when continuously watched. This is why it is also called the ‘watched pot effect’. The effect is the result of repeated, frequent measurements on the system, each measurement projecting the system back to its initial state. In other words, the wave function of the system must repeatedly collapse. It is also necessary that the time interval between successive measurements must be much shorter than the critical time of coherent evolution of the system, called the Zeno time [152]. For decays this Zeno time is the time of coherent evolution before the irreversible exponential decay sets in, and is governed by the reciprocal of the range of energies accessible to the decay products. For most decays this is extremely short and hard to detect [153]. In the case of non-exponential time evolution, the relevant Zeno time can be much longer.

I always thought I would write a book and this is it. In the end, though, I hardly wrote it at all, it evolved from my research notes, from essays I wrote for postgraduates starting work with me, and from lecture handouts I distribute to students taking the relativistic quantum mechanics option in the Physics department at Keele University. Therefore the early chapters of this book discuss pure relativistic quantum mechanics and the later chapters discuss applications of relevance in condensed matter physics. This book, then, is written with an audience ranging from advanced students to professional researchers in mind. I wrote it because anyone aiming to do research in relativistic quantum theory applied to condensed matter has to pull together information from a wide range of sources using different conventions, notation and units, which can lead to a lot of confusion (I speak from experience). Most relativistic quantum mechanics books, it seems to me, are directed towards quantum field theory and particle physics, not condensed matter physics, and many start off at too advanced a level for present day physics graduates from a British university. Therefore, I have tried to start at a sufficiently elementary level, and have used the SI system of units throughout.