The Generalized Representation of Particle Localization in Quantum Mechanics

Kornél Sailer, Zoltán Péli, S. Nagy

A projection method is proposed to treat the one-dimensional Schr\"odinger equation for a single particle when the generalized uncertainty principle (GUP) generates an ultraviolet (UV) wave-vector cutoff. The existence of a unique coordinate representation called the naive one is derived from the one-parameter family of discrete coordinate representations. In this bandlimited quantum mechanics a continuous potential is reconstructed from discrete sampled values observed by means of a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the center and the spreading time of a Gaussian wave packet moving in free space. Indication is found that GUP accompanied by bandlimitation may cause departures of the low-lying energy levels of a particle in a box from those in ordinary quantum mechanics to be much less suppressed than commonly thought when GUP without bandlimitation is at work.

Marc-Thierry Jaekel

The stochastic representation of quantum mechanics is used to give a purely stochastic treatment of the quantum Langevin equation. A general coupling involving positions and velocities of the particle and harmonic modes, with arbitrary dependence on the particle's position, is dealt with. Summing over the modes in the path integral and using stochastic integrals, the author obtains explicitly the reduced measure, that is, the stochastic process representing the time evolution of the small system in the presence of noise. Contact is made with the operatorial representation, and a relation is established between the correlations of the noise in both representations. A general condition on the coupling constants is given for recovering time locality in the continuum limit, and the effects of renormalisation are re-analysed within this purely stochastic approach.

Richard Healey

Jianping Dong, Han Geng

Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials.

Holger Neumann