Two Dimensional Quantum Mechanics on the Noncommutative Space

S. Chaturvedi, R. Jagannathan, V. Srinivasan +1 more

Fan Hong‐Yi, Xubing Tang, Tongtong Wang

For the first time we construct the eigenstate |τ⟩ of noncommutative coordinate. It turns out that |τ⟩ is an entangled state in the ordinary space. Remarkably, its Schmidt decomposition has definite expression in the coordinate representation and the momentum representation. The ⟨τ| representation can simplify some calculations for obtaining energy level of two-dimensional oscillator in noncommutative space.

Won Sang Chung, Eun Ji Jang

S A Alavi

We study the hyperfine splitting in the framework of the noncommutative quantum mechanics (NCQM) developed in the literature. The results show deviations from the usual quantum mechanics. We show that the energy difference between two excited F = I + 1/2 and the ground F = I - 1/2 states in a noncommutative space (NCS) is bigger than the one in the commutative case, so the radiation wavelength in NCSs must be shorter than the radiation wavelength in commutative spaces. We also find an upper bound for the noncommutativity parameter.

Pulak Ranjan Giri, P. Roy

Shi‐Dong Liang

The noncommutative space provides a framework to understand phenomena in Planck scale physics. However, there is no any direct experimental evidence to demonstrate the existence of noncommutative space. We propose an experimental scheme based on the Aharonov-Bohm effect in the nano-scale quantum mechanics to probe the phenomena in the noncommutative phase space. By the Seiberg-Witten map, the free electrons of the nano-scale ring in noncommutative phase space can be mapped equivalently to the quantum mechanical (Heisenberg's algebra) phase space with an extra effective magnetic flux. We introduce two variables related to the persistent current in the ring to probe the noncommutative phase space effect. We give a value-independent criterion to detect the existence of the noncommutative phase space. Namely the answer for existence or nonexistence of the noncommutative phase space depends only on the trend of the curves, rather than the values of the observation data. It can be expected to catch the noncommutative phase-space effect by this scheme.