The Gutzwiller Trace Formula for Quantum Systems with Spin

Holger Cartarius, Jörg Main, Günter Wunner

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.

Yong-Cong Chen

Naobumi Honda

Vaios Blatzios, Christopher H Joyner, Sebastian Müller +1 more

Abstract We derive a Gutzwiller-type trace formula for quantum chaotic systems that accounts for both particle spin precession and discrete geometrical symmetries. This formula generalises previous results that were obtained either for systems with spin (Bolte and Keppeler 1998 Phys. Rev. Lett. 81 1987; 1999 Ann. Phys., NY 274 72) or for systems with symmetries (Robbins 1989 Phys. Rev. A 40 2128–36; Lauritzen 1991 Phys. Rev. A 43 603; Seligman and Weidenmüller 1994 J. Phys. A: Math. Gen. 27 7915–23), but not for a combination of both. The derivation requires not only a combination of methodologies for these two settings, but also the treatment of new effects in the form of double groups and spin components of symmetry operations. The resulting trace formula expresses the level density of subspectra associated to irreducible representations of the group of unitary symmetries in terms of periodic orbits in the system’s fundamental domain. We also derive a corresponding expression for the spectral determinant. In a follow-up paper (Blatzios et al 2025 in preparation) we will show that our formula allows to study the impact of geometrical symmetries and spin on spectral statistics.

Derek W. Robinson

B. Mehlig, Michael Wilkinson

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.