The Emergence of Classical Properties from Quantum Mechanics: New Problems from Old

L. E. Ballentine

Enrico G. Beltrametti

Although it is 70 years old, there are still areas of quantum theory which are open to interpretation. This book deals with such interpretational problems, and in particular with the issue of the measurement process. The main theme is whether quantum mechanics can be viewed as a (semantically) complete theory, namely as a theory capable of describing not only the physical object under observation but also the measuring instruments used to detect the properties of that physical object. The problematic aspects arise from the fact that the measuring instruments themselves behave as classical physical systems. Thus, we are faced with the open problem of the transition from the quantum to the classical, or more explicitly, the problem of why the composition of many quantum micro-objects gives rise to classical macro-objects. This book goes carefully, exhaustively, and deeply into the interpretational problems of quantum mechanics induced by the measurement process. The solution of many relevant issues, as well as the problems that remain unsolved, emerge clearly. The discussion itself is self-contained and the technical parts are reader-friendly. This book is of value to graduate students and researchers who have an interest in the physical, mathematical, and philosophical foundations of quantum mechanics.

W. V. Houston

Recent years have shown a revival of interest in the conceptual problem raised by the incorporation of quantum mechanics into physics. The problem is an old one, but the use of both waves and particles as visualizations of electrons has presented it in a new and acute form. Several observations based on the methods of quantum mechanics can be made. (1) The time development of an isolated system is completely determinate, as in classical mechanics. (2) A precisely defined quantum mechanical state is an eigenstate of some operator described in classical terms. The production of such a state requires a suitable large scale apparatus. (3) If the state is not precisely defined as an eigenstate, quantum statistical mechanics must be used. (4) The idea of measurement is of doubtful applicability in the analysis of idealized experiments, since the process of measurement destroys the identity of the object (an electron) whose properties are measured. Quantum mechanics may be viewed strictly as a method of describing, in classical terms, the behavior of large scale apparatus, although the use of visualizations with a properly conditioned intuition is probably essential to progress in the analysis of new phenomena.

Alberto Cordero

R. Blümel, William P. Reinhardt

Einstein (1917) appreciated early on that within the “old” pre-1925/26 quantum mechanics absence of integrability is a serious obstacle for the quantization of classical systems. Therefore, in retrospect not surprisingly, the quantization problem was not adequately solved until the advent of the “new” quantum mechanics by Heisenberg, Born, Jordan and Schrödinger. The new quantum mechanics did not rely at all on the notion of classical paths, and this way, unwittingly, sidestepped the chaos problem. Within the framework of the new theory, any classical system can be quantized, including classically chaotic systems. But while the quantization of integrable systems is straightforward, the quantization of classically chaotic systems, even today, presents a formidable technical challenge. This is especially true for quantization in the semiclassical regime, where the quantum numbers involved are large. In fact, efficient semiclassical quantization rules for chaotic systems were not known until Gutzwiller (1971, 1990) intoduced periodic orbit expansions. Gutzwiller's method is discussed in Section 4.1.3 below. It is important to emphasize here that the existence of chaos in certain classical systems in no way introduces conceptual problems into the framework of modern quantum theory, although, let it be emphasized again, chaos came back with a vengeance from the “old” days of quantum mechanics. Even given all the modern day computer power accessible to the “practitioner” of quantum mechanics, chaos is the ultimate reason for the slow progress in the numerical computation of even moderately excited states in such important, but chaotic problems as the helium atom.

Jeffrey Bub