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arXiv:quant-ph/9804013

To Enjoy the Morning Flower in the Evening -- Where is the Subtlety of Quantum Mechanics?

Guang-jiong Ni

Published Apr 5, 1998• 0 citations
arXivHTMLPDF

Abstract

Why does the $i=\sqrt{-1}$ appear essentially in the quantum mechanics? Why are there operators and noncommutativity (the uncertainty relation) in the quantum mechanics? Why are these two aspects closely related and indivisible? In probing these problems, a new point of view is proposed tentatively.

DOI
10.48550/arxiv.quant-ph/9804013

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Why does the i=−1i=\sqrt{-1}i=−1​ appear essentially in the quantum mechanics? Why are there operators and noncommutativity (the uncertainty relation) in the quantum mechanics? Why are these two aspects closely related and indivisible? In probing these problems, a new point of view is proposed tentatively.
We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators, we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum-uncertainty states, we derive the minimum-uncertainty state equation by the analytic method and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher-dimensional limit of the uncertainty relations and minimum-uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general noncommuting unitary operators is nontrivially upper bounded.
We review Bohr’s reasoning in the Bohr–Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.
The contents of this paper consist of two independent but correlated topics. In Part I the history of nineteenth and twentieth century philosophy of science is traced in an effort to demonstrate the essential phenomenological aspects of the scientific methodology. Part II, on the other hand, is a technical exposition of some foundational aspects of quantum mechanics based on quantum logic. An effort is made to retain the theme that quantum mechanics is largely a phenomenological theory. As a summary, Part III constitutes an attempt to correlate the first two parts and to present tentatively some consequent reflections on the metaphysical significance of the quantum mechanical formalism.