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Birkhäuser Basel eBooks

Non-Relativistic Quantum Theory

Masao Nagasawa

Published Jan 1, 2000• 4 citations
Publisher

Abstract

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.

DOI
10.1007/978-3-0348-8383-2_3

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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.
We have formulated non-relativistic quantum mechanics as a diffusion theory in the preceding chapters, but ignored to some extent the mathematical problems that arise in connection with the singularity of coefficients of diffusion equations, since we have concentrated on clarifying fundamental concepts and mathematical structures. We shall see that our diffusion processes, Schrödinger diffusion processes, must be treated more carefully. We will analyze some typical stationary cases to see the movement of quantum particles.
The therapeutic approach to quantum theory developed here is meant to be applicable not only to non-relativistic quantum mechanics, but to more advanced quantum theories as well. More specifically, it is meant to be applicable to the quantum theories of systems with infinitely many degrees of freedom, quantum field theories in particular. However, the quantum theories of systems with infinitely many degrees of freedom harbour additional philosophical challenges besides those that are already present in non-relativistic quantum mechanics, and it is to be hoped that the approach suggested here may help to clarify and address those challenges as well. The present chapter turns to some aspects of the interpretation of quantum field theory and assesses them in the light of the Rule Perspective. 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.
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.