By Seidenberg A.

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The tangent vector to a point on the curve is l(x, y). 59) C where S = S2 − S1 is the change in action for one transit around the loop. If the loop does not pass through a wave function node, the action is continuous, and the net change around the loop is zero, S = 0. Now assume that the tiny loop C encircles a node in the wave function at the point (x0 , y0 ). The phase of 32 1. 59, the circulation integral is quantized: = S/m = n(2π) h¯ /m. The integer n can be positive or negative, the sign determines the vortex chirality, and the magnitude measures the state of excitation of the vortex.

61) which is referred to as the quantum stationary Hamilton–Jacobi equation, QSHJE. Over 20 years ago, Floyd began developing a novel trajectory approach for bound stationary states. This approach is quite different from the Bohmian-hydrodynamic formulation that has been surveyed in the preceding sections. Floyd deﬁned the energy dependent modiﬁed potential U = V + Q, in which Q is the quantum potential. Floyd pointed out that for a given eigenvalue, there are an inﬁnite number 1.

The purpose of this term is to moderate the effective potential as the particle encounters the disturbance. When added to the trajectory equations of motion, the artiﬁcial viscosity softens the quantum force and prevents nodes from fully forming. 19]. Adaptive dynamics in hybrid algorithms. 15]. They are adaptive in the sense of i¯h 1. Introduction to Quantum Trajectories 21 solving different dynamical equations in different spatial regions. For example, during a certain time interval, the solution of the TDSE in a nodal region can be linked with the solution of the hydrodynamic equations in the node-free regions.

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