Orthogonal jumps of the wavefunction in white-noise potentials

Wilhelm von Waldenfels

Takeyuki Hida

Lajos Diósi

The master equation of the quantum noise theory is derived by means of Feynman's path integral method. We propose an equivalent s ochastic process where the wavefunction satisfies a nonlinear SchrOdinger equation except for random moments at which it shows orthogonal jumps. Since the early works [ 1-3] on the theory of quantum noise, great interest has been shown in quantum-me-chanical systems affected by random external forces and also new aspects have appeared [4-6]. Usually, the quan-tum state of such systems i described by a density operator obeying a deterministic differential equation (master equation). If we attribute a state vector (instead of the density operator) to the given system, then the state vector should be considered as a stochastic variable governed by a certain stochastic process. In the first part of our work we derive the master equation for the density operator by a seemingly new applica-tion of the path integral technique. Then, we construct a stochastic process for the wavefunction of the system, while retaining its equivalence with the density operator formalism. In the final part we shortly discuss favoured features of the above stochastic process proposed for the wavefunction. This work is an anticipation of a more general and lengthy explanation of the stochastic properties of the wave-function in the presence of random forces. Now, we are going to introduce the notion of stochastic potential of white-noise type. Here we shall investigate the effect of gaussian white-noise potentials. For brevity, we restrict ourselves to the case of a single point.like par-ticle moving in one dimension. Let us assume that the potential V(x, t) acting on the particle is a stochastic variable (x, t stand for the coordi-nate and time). We define the probability distribution of V by the following generator functional G [h]:

Enmin Feng

The white noise approach for stochastic Schrdinger equation(SSE) driven by pure jump Levy white noise is presented.The potential of the SSE is proportional to the Wick exponential of pure jump Levy white noise,which describes the physical system with jump.The white noise approach is based on (S)-1-characterization theorem.The SSE is firstly reduced to the ordinary non-random Schrdinger equation(OSE) by Hermite transform,which can be solved by Feynmann-Kac formula.Then the solution of the SSE is obtained by(S)-1-characterization theorem,converting the solution of the OSE to a(S)-1-process.Furthermore,the solution is in L1(υ) in sense of weak distribution under some certain conditions.

Takeyuki Hida