Hydrogen-atom spectrum under a minimal-length hypothesis

Abstract

The energy spectrum of the Coulomb potential with minimal length commutation relations ͓X i , P j ͔ = iប͕␦ ij ͑1+␤P 2 ͒ + ␤ЈP i P j ͖ is determined both numerically and perturbatively for arbitrary values of ␤Ј / ␤ and angular momenta ᐉ. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of the order of a few GeV −1 , weaker than previously claimed. DOI: 10.1103/PhysRevA.72.012104 PACS number͑s͒: 03.65.Ge, 02.40.Gh, 31.15.Md, 32.10.Fn Quantum gravity incorporates Newton's constant as a dimensional parameter that could manifest itself as a minimal length in the system. Recent string theoretic considerations suggest that this length scale might imply an ultravioletinfrared ͑UV-IR͒ correspondence, contrary to the normal perceptions on momentum and spatial separations. Large momenta are now directly tied to large spatial dimensions, which then implies the existence of a minimal length. Earlier studies have focused upon its amelioration of ultraviolet divergences ͓1͔, but did not take into full account the UV-IR correspondence.There are various ways of implementing such an idea, but the simplest is to suppose that coordinates no longer commute in D-dimensional space. This, in turn, leads to a deformation of the canonical commutation relations. In our previous works, we adopted the equivalent hypothesis that the fundamental commutation relations between position and momentum are no longer constant multiples of the identity. In this paper, we report on constraints on the minimal length hypothesis from precision measurements on hydrogenic atoms. This system has a potential that is singular at the origin, and is therefore particularly sensitive to whether there is a fundamental minimal length. Considerations based upon higher-dimensional theories suggest that such lengths may be large ͓2͔.To set the context, we note that if in one dimension we havewhere ␤ is a small parameter, then the resulting uncertainty relation ͑⌬X͒͑⌬P͒ ജ iប͕1+␤͑⌬P͒ 2 ͖ exhibits a form of the UV-IR correspondence, and gives as minimal length ⌬X ജប ͱ ␤ ͓3͔.We had examined the harmonic oscillator system under this hypothesis in ͓4͔, but no real constraint can be obtained on the minimal length, presumably because of the softness of the potential at the origin. An interesting approach is to take the classical limit ប → 0 of the commutation relations; it yields an unbelievably strong bound, but its robustness might be questioned ͓5͔.We will work in arbitrary D Ͼ 1 dimensions, where ͑1͒ takes the tensorial formwhich, assuming that the momenta commute ͓P i , P j ͔ =0, leads via the Jacobi identity to the nontrivial position commutation relationsThe position and momentum operators can be represented bywhere the operators x i and p j satisfy the canonical commutation relations ͓x i , p j ͔ = iប␦ ij . The simplest representation is momentum diagonal,In this representation the eigenvalue equation for the distance squared operator R 2 = X i X i can be solved exactly. With z = ͑␤ + ␤Ј͒p 2 − 1 ͑␤ + ␤Ј͒p 2 + 1 , ͑5͒ the eig...

Hydrogen atom in momentum space with a minimal length

“…[16][17][18][19][20], where an upper bound for the minimal length was found to be about 0.01 − 0.1 fm. The estimation of this bound was mainly obtained by two methods.…”

“…The problem of the hydrogen atom has been reconsidered by Benczik et al [18], by using the representation given by Eq. (6) Finally, Stetsko and Tkachuk [19] and [20] proposed another perturbative method by using the following position representation of the operators X i and P i :…”

“…(15), the condition η = n, which is not sufficient to assure that the wave function be single valued. In the perturbative treatment of the hydrogen atom [17][18][19][20], β is the perturbation parameter, and, naturally the first-order correction is proportional to this deformation parameter.…”

“…I, several papers have been devoted to the study of the hydrogen atom problem in quantum mechanics with a generalized uncertainty relation, based on the following deformed Heisenberg algebra [16][17][18][19][20]:…”

“…However, this is not often possible, especially when the potential depends, in a not too straightforward manner, on the position operators as in the case of the hydrogen atom potential. In the literature, this problem is the most studied in this modified version of quantum mechanics [16][17][18][19][20]. This is natural because this system has a particular interest.…”