Chapter IV The Axiomatic Structure of Quantum Mechanics

Günter Dähn

Honors Thesis

Masao Nagasawa

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.

P. Enders

This chapter deals with the all-important notion of state as it is central to all parts of physics. For this book, most important are, on the one hand, the notions of state by Newton and Euler – on the other hand, the notion of state by Lagrange and Laplace used nowadays. Surprisingly enough, Newton and Euler’s notions of state are closer to quantum mechanics than the nowadays’ one. Newton’s axiomatic contains solely the conservation of states (1st and 3rd axioms) and the change of states (2nd axiom). In contrast, the equation of motion is not part of the axiomatic. Euler’s axiomatic contains solely the conservation of states. The change of state (and subsequently the equation of motion) is to be examined according to the problem under consideration. In contrast to Newton’s axiomatic and correcting Bohr’s corresponding claim, Euler’s principles of state change for classical bodies are formulated such, that they can be translated even to quantum-mechanical motion. Their power is also demonstrated through an alternative derivation of Hamilton’s equation of motion.

Peter Stolz