Quantum Mechanics of Non-Hamiltonian and Dissipative Systems

M. Moshińsky, Dieter Schuch, Alejandro Suárez-Moreno

The translation from classical to quantum mechanics using the hamiltonian formalism for a non-dissipative systems is very well known. If in the classical picture there is dissipation, the equation of motion is given in terms of newtonian mechanics whose quantum equivalent is not obvious. Using results of previous works we show a way of obtaining a corresponding hamiltonian and discuss the wave function in physical space. We then proceed to analyze the quantum mechanical motion of gaussian wave packets with dissipation.

Kalyan B. Sinha

M. Moshińsky, Dieter Schuch

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.

Antony Streklas

The problem of the dissipative systems from the quantum point of view has been of increasing interest, but it is far from having a satisfactory solution (1). There are many problems where the dissipation has an important role such as in quantum optics, in quantum analysis of fields, in quantum gravity (2). Dissipation can be observed in interactions between two systems, the observed system and another one often called reservoir or the bath, into which the energy flows via an irreversiblemanner (3). The system is embedded in some environmentwhich is in principle unknown. For an effective description of such systems, we can use time dependent Hamiltonians which in classical physics yield the proper equation of motion. If the friction is a linear funcion of the velocity with friction constant γ the Hamiltonian is the well known Caldirola Kanai Hamiltonian

Yuli V. Nazarov, Jeroen Danon