Moyal dynamics and trajectories
201018 pages
Published in:
- J.Phys.A 43 (2010) 025302
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Abstract: (IOP)
We give first an approximation of the operator h: f -> hf := h*planckf -' f*planckh in terms of planck(2)n, n 0, where h\equiv h(p,q), (p,q)\in {\mathbb R}^{2 n} , is a Hamilton function and *planck denotes the star product. The operator, which is the generator of time translations in a *planck-algebra, can be considered as a canonical extension of the Liouville operator L(h): f -> L(h)f := {h, f}(Poisson). Using this operator we investigate the dynamics and trajectories of some examples with a scheme that extends the Hamilton-Jacobi method for classical dynamics to Moyal dynamics. The examples we have chosen are Hamiltonians with a one-dimensional quartic potential and two-dimensional radially symmetric nonrelativistic and relativistic Coulomb potentials, and the Hamiltonian for a Schwarzschild metric. We further state a conjecture concerning an extension of the Bohr-Sommerfeld formula for the calculation of the exact eigenvalues for systems with classically periodic trajectories.- 04.60.Pp
- 03.65.Ta
- 03.65.-w
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