[Paper Review] Minimal Length Uncertainty Relation and gravitational quantum well

Fabian Brau, Fabien Buisseret|Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands|May 18, 2006

Noncommutative and Quantum Gravity Theories2 references3 citations

TL;DR

This paper investigates the gravitational quantum well using a deformed Heisenberg algebra that introduces a minimal length uncertainty, leading to perturbative energy level shifts. Unlike noncommutative geometry approaches, the shifts here are positive and linear in kinetic energy, and comparison with GRANIT experiment data yields a new upper bound of β < 1.46 nm² for the deformation parameter when β is system-dependent, implying a minimal length scale smaller than a few nanometers for neutrons in Earth's gravity.

ABSTRACT

The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance structure characterized by a finite minimal uncertainty in position measurements, a feature it shares with noncommutative theories. We show that an analytical solution can be found in perturbation and we compare our results to those published recently, where noncommutative geometry at the quantum mechanical level was considered. We find that the perturbations of the gravitational quantum well spectrum in these two approaches have different signatures. We also compare our modified energy spectrum to the results obtained with the GRANIT experiment, where the effects of the Earth's gravitational field on quantum states of ultra cold neutrons moving above a mirror are studied. This comparison leads to an upper bound on the minimal length scale induced by the deformed algebra we use. This upper bound is weaker than the one obtained in the context of the hydrogen atom but could still be useful if the deformation parameter of the Heisenberg algebra is not a universal constant but a quantity that depends on the energetic content of the system.

Motivation & Objective

• To examine the effects of a minimal length scale on the energy spectrum of a gravitational quantum well using a deformed Heisenberg algebra.
• To compare the resulting energy shifts with those from noncommutative geometry models and with experimental data from the GRANIT experiment.
• To derive a new upper bound on the minimal length scale when the deformation parameter β is not universal but depends on the system's energetic content.
• To distinguish between different quantum gravity-inspired modifications of the Heisenberg algebra through their unique spectral signatures in a realizable physical system.

Proposed method

• Adopt a two-dimensional deformed Heisenberg algebra with commutation relations modified by a small parameter β, leading to a finite minimal uncertainty in position.
• Use perturbation theory to analytically compute the energy spectrum of a particle in a gravitational potential well under this deformed algebra.
• Derive the energy shift ΔEn as a function of the quantum number n, mass m, and deformation parameter β, showing a linear dependence on kinetic energy.
• Compare the predicted energy shifts with the GRANIT experiment's measured energy level spacings, which resolve transitions with a precision of ~10⁻² peV.
• Determine an upper bound on the minimal length Δx₀ by requiring that the predicted shifts remain below experimental resolution.
• Assess two scenarios: β as a universal constant (using hydrogen atom bounds) and β as a system-dependent parameter (using GRANIT data to derive a new bound).

Experimental results

Research questions

• RQ1 How does the deformed Heisenberg algebra with a minimal length uncertainty modify the energy spectrum of a gravitational quantum well?
• RQ2 What is the signature of the energy level shifts in this model compared to noncommutative geometry approaches?
• RQ3 Can the GRANIT experiment's high-precision data on ultra-cold neutrons in Earth's gravitational field be used to constrain the deformation parameter β?
• RQ4 What upper bound can be placed on the minimal length scale Δx₀ if β is not a universal constant but depends on the system's mass and energy?
• RQ5 How do the perturbative energy shifts from this model compare quantitatively with experimental uncertainties in the GRANIT experiment?

Key findings

• The energy level shifts in the gravitational quantum well due to the deformed algebra are positive and linearly dependent on the kinetic energy of the particle.
• The spectral signature of this model differs from noncommutative geometry models, where shifts can be positive or negative and depend on the square root of kinetic energy.
• If β is a universal constant, the predicted energy shifts are ~10⁻¹⁹ peV, which is far below the GRANIT experiment's resolution of ~10⁻² peV.
• When β is allowed to depend on the system's energetic content, a new upper bound is derived: β < 1.46 nm², corresponding to a minimal length scale Δx₀ < 0.012 eV⁻¹ or ~2.41 nm.
• This new bound is weaker than the one from hydrogen atom spectroscopy but could constrain models where β depends on particle mass or interaction strength.
• The results suggest that the gravitational quantum well is a viable system to distinguish between different quantum gravity-inspired modifications of the Heisenberg algebra.