We find the coordinate space wave functions, maximal localization states, and quasiposition wave functions in a GUP framework that implies a minimal length uncertainty using a formally self-adjoint representation. We show how the boundary conditions in quasiposition space can be exactly determined from the boundary conditions in coordinate space.

The existence of a minimal length uncertainty proportional to the Planck length

It is pointed out by Mead that gravity amplifies the Heisenberg’s measurement uncertainty which makes it impossible to measure distances more accurate than Planck’s length [

The thought experiments that support the minimal length proposal include the Heisenberg microscope with Newtonian gravity and its relativistic counterpart [

Based on the Heisenberg’s microscope and taking into account both the normal and the gravitational uncertainties one finds [

Recently, an experimental scheme is suggested by Pikovski et al. to test the presence of the minimal length scale in the context of quantum optics [

In this paper, we consider a GUP that implies a minimal length uncertainty proportional to the Planck length. We find the exact coordinate space wave functions and quasiposition space wave functions using a formally self-adjoint representation. We first obtain the eigenfunctions of the position operator and the maximal localization states. Then we discuss how the boundary conditions can be imposed consistently in both coordinate space and quasiposition space.

Consider the following one-dimensional deformed commutation relation [

The operator

In this representation, the completeness relation and scalar product can be written as

In this section, we discuss how the boundary conditions in quasiposition space can be determined by fixing the boundary conditions in coordinate space.

Consider the following Dirichlet boundary condition in coordinate space

The Neumann boundary condition determines the values that the derivative of a wave function is to take on the boundary of the domain. Let us consider the following boundary condition in coordinate space:

Now let us elaborate on the correspondence between the uncertainties in position and the imposition of localized boundary conditions. In the GUP framework, it is not possible to measure the position of a particle more accurate than

In this paper, we have investigated the issue of the boundary conditions in deformed quantum mechanics which implies a minimal length uncertainty proportional to the Planck length. We found the coordinate space wave functions, maximal localization states, and quasiposition wave functions using a formally self-adjoint representation. We indicated that the position operator

The authors would like to thank the referees for invaluable comments and important suggestions which considerably improved the quality of the paper.