to Gutzwiller’s trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic field and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement. PACS numbers: 03.65.Sq, 05.45.Mt Keywords: semiclassical theories, Gutzwiller’s trace formula, � expansions, discrete symmetries, diamagnetic hydrogen atom 1.
In the reference (B. Gremaud, Phys. Rev. E 65, 056207 (2002)) flrst-order ~ corrections to Gutzwiller's trace formula for systems with a smooth potential were presented. Here we present an extension of that theory to systems with discrete symmetries. We apply the method to the two-dimensional hydrogen atom in a uniform magnetic fleld and exploit the C4v-symmetry in the calculation of the correction terms. The numerical results for the semiclassical values are compared with values extracted from exact quantum mechanical calculations. The comparison shows a very good agreement.
We study the quantum propagator in the semiclassical limit with hard-wall potentials. We show that, upon each reflection by the hard wall, a Berry phase
π is accumulated and leads to interferences between different classical trajectories. Including the Berry phase caused by the hard walls, the modified Van Vleck's formula is derived. We also discuss the close relations among quantum statistics, discrete gauge symmetry, and hard-wall constraints. Most of all, we formulate a new quantization rule that applies to {\it both} smooth and sharp boundary potentials. It provides an easy way to compute quantized energies in the semiclassical limit and is extremely useful for many physical systems.