A symmetry trip from Caldirola to Bateman damped systems

Details A symmetry trip from Caldirola to Bateman damped systems A symmetry trip from Caldirola to Bateman damped systems

2011-02-04

arxiv.org/abs/1102.0990v1

Physics

Mathematical Physics

16 pages

Scientific paper

For the Caldirola-Kanai system, describing a quantum damped harmonic oscillator, a couple of constant-of-motion operators generating the Heisenberg algebra can be found. The inclusion of the standard time evolution symmetry in this algebra for damped systems, in a unitary manner, requires a non-trivial extension of this basic algebra and hence the physical system itself. Surprisingly, this extension leads directly to the so-called Bateman's dual system, which now includes a new particle acting as an energy reservoir. The group of symmetries of the dual system is presented, as well as a quantization that implies, in particular, a first-order Schr\"odinger equation. The usual second-order equation and the inclusion of the original Caldirola-Kanai model in Bateman's system are also discussed.

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