Particle in a cavity in one-dimensional bandlimited quantum mechanics

Abstract

The effects of the generalized uncertainty principle (GUP) on the low-energy stationary states of a particle moving in a cavity with no sharp boundaries are determined by means of the perturbation expansion in the framework of onedimensional bandlimited quantum mechanics. A realization of GUP resulting in the existence of a finite ultraviolet (UV) wave-vector cutoff ∼ K ℓ 1 P (with the Planck length ℓ P ) is considered. The cavity of the size ≫ ℓ ℓ P is represented by an infinitely deep trapezoid-well potential with boundaries smeared out in a range R satisfying the inequalities ≫ ≳ ℓ R ℓ P . In order to determine the energy shifts of the low-lying stationary states, the usual perturbation expansion is reformulated in a manner that enables one to treat consistently order-by-order the direct and indirect GUP effects, i.e., those due to the modification of the Hamiltonian and the lack of the UV modes, respectively. It is shown that the leading terms of the indirect and the direct GUP effects are of the first and second order, respectively, in the small parameter ℓ ℓ P in agreement with our previous finding in a more naive approach Sailer et al (2013 Phys. Rev. D 87 084056).

Non-local imprints of gravity on quantum theory

“…The general reasoning so far given can readily be compared with the momentum cut-off approach for implementing the concept of minimum length into QM [11,12]. This approach implies to restrict the Hilbert space of state vectors to the cut-off functions…”

“…Let us note in passing that the corrections to the low-lying energy levels can be found by exploiting the standard perturbation theory, see section II D. On the other hand, for energy levels which are high enough (close to ß 2 /2m) one may neglect the second derivative in Eq. (12). Under this assumption, one arrives at the equation…”

“…Let us briefly summarize the key points concerning band-limited QM. As it was suggested in [11,12], by using the projection operator (4), one can easily find the modified Schrödinger equation compatible with the momentum cutoff of the wave function. But the problem of compatibility still persists as far as we are concerned with the boundary conditions.…”

“…In view of the sampling theorem in information theory [9], which tells one how to digitize an analog signal in a precise way, it was noticed in [10] that the application of hard momentum cutoff to the fields implies their representation on the lattice and thus may be considered as one of the simplest ways for introducing discrete space. Inspired by this idea, some basic features of QM with hard momentum cutoff has been worked out in [11,12]. So far fairly little attention has been paid to this discussion.…”

IR/UV corrections to the Casimir force

“…Loosely speaking, The implementation of minimum length in quantum mechanics can be done either by modification of position and momentum operators or by the restriction of their domains. The latter possibility (see for example [18,19]) is somewhat advantageous over the minimum-length deformation of Weyl-Heisenberg algebra [20] as in the latter case one faces unacceptably large effects in a classical limit [21,22]. In general, such theories imply modified dispersion relation and momentum cutoff set by the quantum gravity scale.…”