On the quantum mechanical treatment of dissipative systems

Abstract

Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.

Quantum Wave Equation of Non-conservative System

“…The nonconservative quantum processes are common too, since there is dissipation in every nonequilibrium or fluctuating process, including tunneling [17][18][19], electromagnetic cavity radiation [20,21], masers and parametric amplification [21], Brownian motion [14,22,23], inelastic scattering [15,24], squeezed states of quantum optics [25], and electrical resistance or Ohmic friction [26,27]. In the experiment of Bose-Einstein condensates, the atomic Rb 87 , Na 23 and Li 7 can be cooled in laser field, since they get the non-conservative force from the photons, and the force F = −k v. Substituting F = −k v into (16), we have…”

“…In Refs. [14,15], Schrödinger's equation was modified directly. For example, a nonlinear term proportional to ln(ψ/ψ * ) can be added, and these types of modifications provide quantum results corresponding to classical friction.…”

Quantization of non-Hamiltonian and dissipative systems

“…To quantize the operator Q k 1,2 we use formulas (12), (13). It is known [25,24] that Weyl quantization (12), (13) ,p).…”

“…It is known [25,24] that Weyl quantization (12), (13) ,p). Therefore, Weyl quantization of the operators Q k 1,2 lead to superoperatorŝ…”

Quantum Analysis on Time Behavior of a Lengthening Pendulum

“…In fact, the description of quantized energy for a TDHS, such as (10), is a delicate problem. While some authors [1,3,11,21,22] including ours have investigated quantized energy levels for specific TDHSs, there is another opinion [26,27] that such energy levels do not exist for the case of TDHSs. Hence, explicit demonstrations of the existence of quantized energy levels may be an interesting research topic for further study in the future in this field.…”