Investigation of Free Particle Propagator with Generalized Uncertainty Problem

We consider the Schrodinger equation with a generalized uncertainty principle for a free particle. We then transform the problem into a second ordinary differential equation and thereby obtain the corresponding propagator. The result of ordinary quantum mechanics is recovered for vanishing minimal length parameter.


Introduction
The generalization of classical action principle into quantum theory appears in path integral formulation. Instead of a single classical path, the quantum version considers a sum, or better say, integral of infinite possible paths [1,2]. Although the main idea of path integral approach was released by N. Wiener, in an attempt to solve diffusion and Brownian problems, it was introduced in Lagrangian formulation of quantum mechanics of P. A. M. Dirac [1,2]. Nevertheless, the present comprehensive formulation is named after Feynman and extracted from his PhD thesis supervised by J. A. Wheeler [1,2]. Feynman's formulation is now an essential ingredient in many fundamental theories of theoretical physics including quantum field theory, quantum gravity, high energy physics, etc [1,2,3].
On the other hand, we are now almost sure from fundamental theories such as string theory and quantum gravity that the ordinary quantum mechanics ought to be reformulated. In more precise words, a generalization of Heisenberg uncertainty principle, called Generalized Uncertainty Principle (GUP), should be considered at energies of order Planck scale [4][5][6][7]. This generalization corresponds to a generalization of wave equation of quantum mechanics. Till now, various wave equations of quantum mechanics, different interactions, as well as other related mathematical aspects and physical concepts have been considered in this framework [8][9][10][11][12][13][14][15][16][17][18].
In our note, we are going to combine these two subjects. Namely, we study the free particle propagator in Schrödinger framework in minimal length formulation. In section 2, we review the essential concepts of GUP and write the generalized Hamiltonian for free particle. In section 3, we obtain propagator for this system that the details of calculations are brought.

GUP-Corrected Hamiltonian
An immediate consequence of the ML is the GUP where the GUP parameter α is determined from a fundamental theory. At low energies, i.e. energies much smaller than the Planck mass, the second term in the right hand side of Eq. (1) vanishes and we recover the well-known Heisenberg uncertainty principle. The GUP of Eq. (1) corresponds to the generalized commutation relation It should be noted that in the deformed Schrödinger equation, the Hamiltonian does not have any explicit time dependence [19]; This deformed momentum operator modifies the original Hamiltonian as The problem becomes much simpler if we consider [16]; In the free particle case, we therefore have We now calculate the single free particle propagator corresponding to this deformed Hamiltonian in section 3.

Perspicuous Form of Propagator
If the wave function ( , ') x t  is known at a time ' t , we can explicitly write the wave function '' ( , ) x t  at a later time " t using the propagation relation as [20]   In more explicit form, the propagator for free particle under minimal length is Now, if we assume 0   , then the one dimensional free particle propagator is given by In order to obtain free particle propagator for a finite time interval   (16), the propagator is obtained as Replacing Now, if we calculate the probability of detecting the particle at a finite region  which is the result in ordinary quantum mechanics.