Topological Effects in Quantum Mechanics

G N Afanasiev

Akira Inomata

J. Tolar

Hasan Özgür Çıldıroğlu, A. U. Yilmazer

The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.

H. D. Doebner, H. J. Elmers, W. Heidenreich

Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.

Jincheng Zhang

Quantum Topologies, a rapidly developing field of research, explores the use of quantum mechanics to model and optimize complex geometric shapes. This paper investigates the application of quantum topology constraints to solve problems with intricate geometric requirements, aiming to overcome limitations inherent in classical optimization techniques. We propose a novel algorithm – Quantum Topologies Constrained Optimization (QTO) – that utilizes quantum topology to guide the optimization process, enhancing solution quality and efficiency. The core mechanism centers around the inherent non-linearity of topology, which allows for novel exploration of the solution space, potentially leading to more effective and robust optimization. The paper presents a detailed theoretical framework and a practical implementation demonstrating the effectiveness of QTO in solving challenging geometric constraints.