We show how Wegner's flow equations can be reformulated as ordinary differential equations through the use of the Moyal bracket. In finite-dimensional Hilbert spaces the introduction of the Moyal bracket leads naturally to the identification of a small expansion parameter, namely the inverse of the dimensionality of the space. This expansion corresponds to a non-perturbative treatment of the coupling constant. In the case of infinite-dimensional spaces ℏ plays the role of the small parameter and the Moyal formulation then allows for a semi-classical treatment of the flow equation. We demonstrate these statements for the Lipkin and Dicke models as well as for the symmetric x4 and double-well potentials.
Non-commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non-commutative configuration space. Within this framework an unambiguous definition can be given for the non-commutative well. Using this approach we compute the bound state energies, phase shifts and scattering cross sections of the noncommutative well. As expected the results are very close to the commutative results when the well is large or the non-commutative parameter is small. However, the convergence is not uniform and phase shifts at certain energies exhibit a much stronger then expected dependence on the non-commutative parameter even at small values.
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