20. Integral Equations in Quantum Mechanics

Chandra Halim, Muhammad Farchani Rosyid

The implementation of Lévy path integral generated by Lévy stochastic process on fractional Schrödinger equation has been investigated in the framework of fractional quantum mechanics. As the comparison, the implementation of Feynmann path integral generated by Wiener stochastic process on Schrödinger equation also has been investigated in the framework of standard quantum mechanics. There are two stochastic processes. There are Lévy stochastic and Wiener stochastic process. Both of them are able to produce fractal. In fractal’s concept, there is a value known as fractal dimension. The implementation of fractal dimension is the diffusion equation obtained by using Fokker Planck equation. In this paper, Lévy and Wiener fractal dimension have been obtained. There are for Lévy and 2 for Wiener/Brown fractal dimension. Fractional quantum mechanics is generalization of standard quantum mechanics. A fractional quantum mechanics state is represented by wave function from fractional Schrödinger equation. Fractional Schrödinger equation is obtained by using kernel of Lévy path integral generated by Lévy stochastic process. Otherwise, standard quantum mechanics state is represented by wave function from standard Schrödinger equation. Standard Schrödinger equation is obtained by using kernel of Feynmann path integral generated by Wiener/Brown stochastic process. Both Lévy and Feynmann Kernel have been investigated and the outputs are the Fourier Integral momentum phase of those kernels. We find that the forms of those kernels have similiraty. Therefore, we obtain Schrödinger equation from Lévy and Feynmann Kernel and also the comparison of Lévy energy in fractional quantum mechanics and particle energy in standard quantum mechanics.

Tal Hendel

Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

Vladimir V. Ulyanov

D. Sokolovski

The quantum-mechanical traversal- (tunneling) time problem is reformulated in terms of a square-integrable amplitude distribution (wave function) whose evolution is governed by a generalized Schr\"odinger equation. The method uses an analogy between quantum mechanics and classical Brownian motion. Brownian and quantum traversal-time distributions are evaluated by the direct integration of corresponding equations of motion. The general relation between Feynman and Schr\"odinger quantum mechanics is discussed.