Mathematical Concepts of Quantum Mechanics

Stephen J. Gustafson, Israel Michael Sigal

D. Morris

Stephen J. Gustafson, Israel Michael Sigal

F. J. Duarte

Quantum mechanics is based on the concepts of indeterminacy , indistinguishability probability amplitudes , probabilities , reversibility , superposition , and uncertainty . And yet that is only part of the essence of quantum mechanics. Using the words attached to the concepts just described provides only a partial description of what quantum mechanics really is. The other part is provided by a set of physical and mathematical principles annexed to the mentioned concepts. The beauty is that quantum mechanics becomes workable and functional just by observing its mathematical principles. There is no need for any particular interpretation. Not even the orthodox interpretation. Here, fifteen mathematical-physical principles of quantum mechanics are listed and succinctly explained.

Kong Lingfang

Abstract For a long time, the academic community believes that quantum mechanics and classical mechanics are two greatly different mechanical systems, and the algorithms of quantum mechanics cannot be used to solve classical mechanics problems [1] . However, starting from classical mechanics, this paper proves through rigorous mathematical methods that classical mechanics and quantum mechanics have exactly the same mathematical algorithms. For example, concepts such as wave function, probability amplitude and superposition state can also describe the random motion states of macroscopic objects. This paper mainly adopts mathematical calculation, supplemented by computer simulation experiments, and verifies the correctness of the mathematical calculation process from both theoretical and experimental perspectives. No assumptions are needed in this paper, and all concepts are derived and proved through strict mathematical deduction. These derivation steps show that concepts such as probability amplitude and wave function are universal concepts applicable to random motion processes, which can be used for both macroscopic objects and microscopic particles, and are not unique calculation methods of quantum mechanics. PACS: 03.65.Ta, 03.50.Kk

M. A. Bashar, Mahmudul Hasan, M. Syduzzaman M. A. Chowdhury

The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.Mathematics lies at the heart of quantum computation and it is needed in every aspect of the research and development of quantum computation.In this paper we have presented mathematical concepts that are closely related to quantum computation and their relation to quantum mechanics in a way which is innovative and very easy to comprehend.Most of the mathematical concepts presented in this paper are deduced from the postulates of quantum mechanics and linear algebra.