Quantum Mechanics in a 64-DIMENSIONAL Universe.

Ying Hu, Zhaoxin Liang

This paper gives a systematic review on studies of dimensional effects in pure- and quasi-two-dimensional (2D) Bose gases, focusing on the role of dimensionality in the fundamental relation among the universal behavior of breathing mode, scale invariance and dynamic symmetry. First, we illustrate the emergence of universal breathing mode in the case of pure 2D Bose gases, and elaborate on its connection with the scale invariance of the Hamiltonian and the hidden SO(2, 1) symmetry. Next, we proceed to quasi-2D Bose gases, where excitations are frozen in one direction and the scattering behavior exhibits a 3D to 2D crossover. We show that the original SO(2, 1) symmetry is broken by arbitrarily small 2D effects in scattering, which consequently shifts the breathing mode from the universal frequency. The predicted shift rises significantly from the order of 0.5% to more than 5% in transiting from the 3D-scattering to the 2D-scattering regime. Observing this dimensional effect directly would present an important step in revealing the interplay between dimensionality and quantum fluctuations in quasi-2D.

Diederik Aerts, Massimiliano Sassoli de Bianchi

We show that the extended Bloch representation of quantum mechanics also applies to infinite-dimensional entities, to the extent that the number of (possibly infinitely degenerate) outcomes of a measurement remains finite, which is always the case in practical situations.

Mikio Namiki, Y. Aizawa, Kei‐ichi Maeda +1 more

Peter S. Buffa, Sally V. Beaty, Annenberg +1 more

Mukhtarov Erkin Kobilzhonovich

It is impossible for students to master the basic ideas and conclusions of quantum mechanics without solving a certain set of problems. Problems play a significant role in the educational process, forming not only logical thinking, but also influencing the overall development of the students. The article poses and solves problems about stationary one-dimensional problems in quantum mechanics.