# One-electron atoms in Schwarzschild universe: bare and electromagnetically dressed cases

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## Abstract

The quantum mechanics of one-electron atoms in the presence of external electromagnetic fields is considered within Weber’s framework. The results by the earlier studies are extended in the sense that for given source and field configurations the changes of the electromagnetic potentials due to the curved background are included. The formulation is specialized to the case with Schwarzschild background. The first corrections to the energy levels for bare atom and Zeeman/Stark effects are calculated, exhibiting possible changes in meaningful orders.

## Keywords

Test Particle Stark Effect Zeeman Effect Curve Background Uniform Electric Field## 1 Introduction

The behavior of quantum mechanical systems in the presence of gravitational fields has been the subject of a great number of research publications. Among others, two leading approaches are those by DeWitt [1] and Weber [2]. In the studies based on DeWitt’s approach, the general formulation of quantum mechanics for a relativistic or nonrelativistic system on a curved background is the main concern [3, 4, 5, 6]. In the latter, based on Weber’s, an interaction scheme between the quantum system and the gravitational field is the guiding rule. In particular, in this approach the linearized classical equations of motion of the test system/particle interacting with the gravitational fields provide the basic ingredients to formulate the quantum theory [2, 7, 8]. Interestingly, these two approaches are not equivalent, and they are based on different sequences and orders of approximations being used in each approach; one may get different results [7, 8].

Based on the DeWitt approach, the formulation of Dirac particles on a curved background is used to extract the first corrections in curvature to the energy levels of one-electron atoms [3, 4, 5, 6]. In [7, 8], the interaction between gravitational waves and a charged test particle is studied. While [7, 8] falls within Weber’s scheme, it is shown that the sequence of linearizations used in the original version [2] is not sufficient in the case that one is dealing with the charged particles in the presence of external fields.

The purpose of the present work is to extend the results for nonrelativistic charged particles on a curved background. In particular, within Weber’s framework, we consider the case with one-electron atoms in the presence of additional external electromagnetic fields in the small curvature limit to obtain the first corrections to the energy levels. Extending the results by [3, 4, 5], for given source or field configurations, the corrections due to curvature to the electromagnetic potentials as well as and their effects on the energy levels are studied. It will be seen that the obtained corrections to the nuclei potential and the external fields due to curvature can result in changes in the meaningful orders of magnitude. As a specific example, the corrections to the energy levels of the one-electron atom in the Schwarzschild metric is considered.

The scheme of the rest of this paper is the following. In Sect. 2, the basic notions of the formulation on curved background, including the Riemann normal coordinate system is reviewed. In Sect. 3, the basic elements of the quantization procedure as well as the construction of the Hamiltonian in the presence of the electromagnetic potentials based on Weber’s approach are presented. In Sect. 4 the formulation is specialized to the case of a one-electron atom in a Schwarzschild background. In particular, for the case of a bare atom and the Zeeman/Stark effects the first corrections to the energy levels are obtained. Section 5 is devoted to our concluding remarks.

## 2 Basic notions

*Riemann normal coordinate system*[2, 9, 10, 11, 12]. The metric components have the following forms in the Riemann coordinates up to the first order of Riemann’s tensor (\(i,j,\cdots =1,2,3\)):

## 3 Toward quantum system

Here, using a set of assumptions and approximations, we develop the quantum mechanics governing the dynamics of the test particle. As announced earlier, our approach is basically the one by Weber’s.

## 4 Quantum theory in Schwarzschild background

### 4.1 Bare one-electron atom

### 4.2 The normal Zeeman effect

### 4.3 The Stark effect

## 5 Concluding remarks

The results for nonrelativistic charged particles on a curved background are extended. In particular, within the Weber framework, we consider the case with one-electron atoms in the presence of additional external electromagnetic fields in the small curvature limit to obtain the first corrections to the energy levels. Extending the results by [3, 4, 5], for given source or field configurations, the corrections due to curvature to the electromagnetic potentials as well as their effects on the energy levels are studied. It is seen that the obtained corrections to the nuclei potential and the external fields due to curvature can result in changes in meaningful orders of magnitude. As a specific example, the corrections to the energy levels of the one-electron atom in the Schwarzschild metric is considered. In particular, for the case with a bare atom it is observed for lower values of quantum number \(n\) that the corrections to the scalar potential of the nucleus cannot be ignored and are comparable with the corrections by [5]. In the case of the Zeeman effect it is seen that, as is well known, the Hamiltonian would take a different form in comparison with that based on DeWitt’s approach [5]. As a consequence, the semi-classical behavior of the systems would be different in comparison with a similar treatment of the system in [5].

## Notes

### Acknowledgments

The author is grateful to A. H. Fatollahi for useful comments. Also the author thanks the Shahrekord University for support of this research grant fund.

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