β Scribed by W Stocker; K Albrecht
No coin nor oath required. For personal study only.
We generalize the classical Hamilton-Jacobi equation and Madelung's fluid dynamical formulation of the Schrijdinger equation via a substitution prescription for the phase S of the wavefunction. The time and space derivatives of S are altered as $ -$ + f and s' -+ S' + g. In the classical limit (when S becomes the principal function) this substitution yields an additional force in the equation of motion. To generate a friction force, the functions f(x, t) and g(x, t) must have a certain structure following from the corresponding classical case. They must necessarily depend on the wavefunction itself, which leads to a nonlinear Schrijdinger equation. With given f and g, the quanta1 Hamiltonian follows in a unique way. The Hamiltonians proposed by Kostin and by Siissmann et al. turn out as special examples within a larger family of classically equivalent friction operators. Schriidinger equations for nonlinear frictional motion can also be constructed. The quanta1 energy flow equation is generalized consistently.
π SIMILAR VOLUMES
We present a new method to derive transport equations for non-relativistic quantum manyparticle systems. This method uses an equation-of-motion technique and is applicable to interacting fermions and (or) bosons in arbitrary time-dependent external fields. Using a cluster expansion of the r-particle
classical mechanical system coupled to a heat reservoir through frictional forces is established. The explicit expressions for the diffusion coefficient and the drift velocity which were formulated in preceding articles are evaluated here by an expansion in inverse powers of the frictional constant