•  
  •  
 

Journal of Undergraduate Research

Keywords

non-commutative quantum mechanics, harmonic oscillator

College

Physical and Mathematical Sciences

Department

Physics and Astronomy

Abstract

Non-commutative quantum mechanics is an active field of research in which the operators corresponding to different spatial dimensions are assumed to not commute, as a consequence of quantization of spacetime. This field is important because it investigates the relationships between fundamental physical ideas such as space, time, momentum, and position on very small scales and will provide the basis for experiments probing these small scales in the future. A well-known result from this field of research maps the simple harmonic oscillator (SHO) system in non-commutative coordinates into a corresponding system in normal quantum mechanics. The corresponding system is still a SHO, but with a constant magnetic field superimposed on top of the potential. Results such as these are fascinating and drive the field of non-commutative quantum mechanics to be extremely active. The commutation algebras used are generally not Lorentz-invariant. Since there is no experimental evidence that Lorentz breaking occurs, the domain of applicability of these results is unknown. By contrast, Hartland Snyder published the first formulation of quantum mechanics in a quantized spacetime in 1947 in a way to guarantee Lorentz-invariance. His motivations were quite different from those investigating the field today, and so his formulation is usually only cited for historical reasons. For these reasons, i.e. historical significance and a unique approach to an active field of research, we chose to investigate the simple harmonic oscillator in Snyder Space.

Download

Included in

Physics Commons

Share

COinS