Quantum Confinement is in some sense a new subject. International meetings dedicated to Quantum Confinement have occurred only recently in Mexico City (the first in 2010 and the second, in September 2011). However, at least in principle, the subject has existed since a very long time. Surprisingly perhaps, it lay dormant for many years, for want of suitable experimental examples. However, when one looks carefully at its origin, it turns out to have a long and distinguished history. In fact, the problem of quantum confinement raises a number of very interesting issues concerning boundary conditions in elementary quantum mechanics and how they should be applied to real problems. Some of these issues were missed in the earliest papers, but are implicit in the structure of quantum mechanics, and lead to the notion of Confinement Resonances, the existence of which was predicted theoretically more than ten years ago. Although, for several reasons, these resonances remained elusive for a very long time, they have now been observed experimentally, which puts the whole subject in much better shape and, together with the advent of metallofullerenes, has contributed to its revival.
Quantum Confinement is in some sense a new subject. International meetings dedicated to Quantum Confinement have occurred only recently in Mexico City (the first in 2010 and the second, in September 2011). However, at least in principle, the subject has existed since a very long time. Surprisingly perhaps, it lay dormant for many years, for want of suitable experimental examples. However, when one looks carefully at its origin, it turns out to have a long and distinguished history. In fact, the problem of quantum confinement raises a number of very interesting issues concerning boundary conditions in elementary quantum mechanics and how they should be applied to real problems. Some of these issues were missed in the earliest papers, but are implicit in the structure of quantum mechanics, and lead to the notion of Confinement Resonances, the existence of which was predicted theoretically more than ten years ago. Although, for several reasons, these resonances remained elusive for a very long time, they have now been observed experimentally, which puts the whole subject in much better shape and, together with the advent of metallofullerenes, has contributed to its revival.
It is a surprise that a theory that seems to be in conflict with so much of the real world continues to serve as the underpinning for our understanding of that world. Quantum Mechanics has been with us in its present form since the mid-1920's, and still serves as one of the two basic frameworks on which all meaningful physical theories must rest. The question persists: Can quantum mechanics fail to predict phenomena we can observe with new precision, or does it possibly have limits on its applicability? Or is it an approximation to a more general theory? With the recent advances in experimental techniques and in theoretical thinking, it seemed appropriate to examine these questions anew through the mechanism of an informal conference. The idea for holding a workshop now came from the recent advances in both theoretical studies of quantum mechanics, and in the new classes of experiments that can provide new tests or improve significantly on old ones. The papers in these Proceedings are divided between theoretical studies and discussions of experiments. Most of the theoretical work describes the quantum mechanical analysis of new systems not hitherto analyzed. Half (or more) of the experimental papers describe new tests of the predictions of quantum theory, either by employing new precision on existing experiments or by exploring new ground.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
I. Brief History.When I first learned of this conference, and saw the wide range of interests represented by the participants, I was certain that finally a common definition of that most intriguing concept-the soliton-would emerge.And I was anxious that this definition be broad enough to encompass solitons in both their classical and quantum versions.For recent work had convinced me that the general technique known as Bethe's ansatz for solving diverse one-dimensional quantum problems in fact was nothing more nor less than the quantum soliton.However, from conversations with conference participants, I discovered that many were not aware of this accumulated work on exactly soluble quantum systems.I even gained the impression that some feel quantum mechanics to be much harder than classical mechanics-an unnecessary complication.My own feeling, on the other hand, is that in many cases quantum mechanics clarifies matters.It forces one to immediately face problems that would eventually have to be faced in the corresponding classical case, problems such as the counting of states, to determine how important solitons really are.Anyway, the history of the quantum soliton is all to the greater glory of the soliton concept.And quite a respectable history it is.I would date the beginning at 1931 with a paper of H. Bethe [1] on magnetism; that is a total span of 45 years.Quantum mechanics itself has not been around all that much longer!It now seems my hopes for a consensus on a definition of the soliton were premature, and things are still fermenting.Nevertheless, here is my contribution to the conference in the form of a brief history for general interest, and a more original work on the quantization of the Toda lattice.Perhaps next time there will be some consolidation, and we can then decide just how particular and how general a concept is the soliton.1.We begin this historical account with the classic paper of H. Bethe [1], which introduced first the many-body wave function crucial
A succinct history of the Confined Atom problem is presented. The hydrogen atom confined to the centre of an impenetrable sphere counts amongst the exactly soluble problems of physics, alongside much more noted exact solutions such as Black Body Radiation and the free Hydrogen atom in absence of any radiation field. It shares with them the disadvantage of being an idealisation, while at the same time encapsulating in a simple way particular aspects of physical reality. The problem was first formulated by Sommerfeld and Welker [1]—henceforth cited as SW—in connection with the behaviour of atoms at very high pressures, and the solution was published on the occasion of Pauli's 60th birthday celebration. At the time, it seemed that there was not much other connection with physical reality beyond a few simple aspects connected to the properties of atoms in solids, for which more appropriate models were soon developed. Thus, confined atoms attracted little attention until the advent of the metallofullerene, which provided the first example of a confined atom with properties quite closely related to those originally considered by SW. Since then, the problem has received much more attention, and many more new features of quantum confinement, quantum compression, the quantum Faraday cage, electronic reorganisation, cavity resonances, etc have been described, which are relevant to real systems. Also, a number of other situations have been uncovered experimentally to which quantum confinement is relevant. Thus, studies of the confined atom are now more numerous, and have been extended both in terms of the models used and the systems to which they can be applied. Connections to thermodynamics are explored through the properties of a confined two‐level atom adapted from Einstein's celebrated model, and issues of dynamical screening of electromagnetic radiation by the confining shell are discussed in connection with the Faraday cage produced by a confining conducting shell. The conclusions are shown to be relevant to a proposed 'quantum computer'. The description of the actual geometry of C60, as opposed to a purely spherical approximation, leads to some qualification of the computed results.