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Birkhäuser Basel eBooks

Micro Statistical Theory

Masao Nagasawa

Published Jan 1, 2000• 0 citations

Abstract

When the movement of a particle with noise was introduced in Chapter I, we were interested in the source of the noise, but postponed further discussion about it, accepting that the noise is presumably caused by the structure of the vacuum itself, and also by our mathematical method of treating the problem. We now return to the subject. On the other hand, we have seen in Chapter IV that the single particle interpretation of quantum mechanics is problematic, since in exited states of an electron, say in a hydrogen atom, the ergodic decomposition occurs at the zero sets of distribution densities (see Theorems 5.2.2 and 5.4.2). In other words, since the tunnelling is not allowed at the zero sets, the long-time average of a sample path cannot realize the predicted distribution of an electron in an excited state. This indicates that quantum mechanics must be a statistical theory. To clarify this point we need an additional mathematical structure. A statistical model was discussed in chapters VII and VIII of Nagasawa (1993) to solve this problem. We will explain it briefly, and apply it to relativistic quantum particles. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1007/978-3-0348-8383-2_13

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When the movement of a particle with noise was introduced in Chapter I, we were interested in the source of the noise, but postponed further discussion about it, accepting that the noise is presumably caused by the structure of the vacuum itself, and also by our mathematical method of treating the problem. We now return to the subject. On the other hand, we have seen in Chapter IV that the single particle interpretation of quantum mechanics is problematic, since in exited states of an electron, say in a hydrogen atom, the ergodic decomposition occurs at the zero sets of distribution densities (see Theorems 5.2.2 and 5.4.2). In other words, since the tunnelling is not allowed at the zero sets, the long-time average of a sample path cannot realize the predicted distribution of an electron in an excited state. This indicates that quantum mechanics must be a statistical theory. To clarify this point we need an additional mathematical structure. A statistical model was discussed in chapters VII and VIII of Nagasawa (1993) to solve this problem. We will explain it briefly, and apply it to relativistic quantum particles. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The equation of motion of a quantum particle(s) in a potential field will be introduced. The equation of motion should determine drift fields a (t, x) and the movement X t of quantum particles, accordingly. However, the interrelation between the equation of motion and the movement X t of a particle(s) is not so direct as in classical mechanics, and we must go into deeper mathematical structures of diffusion processes to clarify the relation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.

As explained in the previous section, the currently most-discussed interpretations of quantum theory can be classified according to how they (try to) solve the measurement problem. Accounts which are based on what I call the ‘therapeutic’ type of approach to philosophical problems propose a radically different strategy to overcome the measurement problem. The present chapter presents the essential ideas behind therapeutic approaches, briefly sketches their background in the history of philosophy, and explains how one might try to put them to work in the foundations of quantum theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.