A symmetric treatment of damped harmonic oscillator in extended phase space

Extended phase space (EPS) formulation of quantum statistical mechanics treats the ordinary phase space coordinates on the same footing and thereby permits the definite the canonical momenta conjugate to these coordinates . The extended lagrangian and extended hamiltonian are defined in EPS by the same procedure as one does for ordinary lagrangian and hamiltonian. The combination of ordinary phase space and their conjugate momenta exhibits the evolution of particles and their mirror images together. The resultant evolution equation in EPS for a damped harmonic oscillator, is such that the energy dissipated by the actual oscillator is absorbed in the same rate by the image oscillator leaving the whole system as a conservative system. We use the EPS formalism to obtain the dual hamiltonian of a damped harmonic oscillator, first proposed by Batemann, by a simple extended canonical transformations in the extended phase space. The extended canonical transformations are capable of converting the damped system of actual and image oscillators to an undamped one, and transform the evolution equation into a simple form. The resultant equation is solved and the eigenvalues and eigenfunctions for damped oscillator and its mirror image are obtained. The results are in agreement with those obtained by Bateman. At last, the uncertainty relation are examined for above system.