Is there a stable hydrogen atom in higher dimensions?

Abstract

The Schrödinger equation in higher dimensions is considered. It consists of the kinetic energy part given by the corresponding Laplace operator, and a term describing the interaction with the electrostatic field of a point charge. From Rutherford-type scattering experiments one can conclude that the potential of a point charge is ∼1/r irrespective of the dimension of the space where the experiment is carried through. Also the structure of the kinetic energy is unchanged in higher dimensions so that one is lead to the result that there exist stable atoms in higher spatial dimensions d⩾4. The solutions and energy eigenvalues to this Schrödinger equation in higher dimensions are presented. As a consequence, the dimensionality of space can be read off from the spectral scheme of atoms: The three-dimensionality of space is a consequence of the existence of the Lyman series. Another consequence is that the Maxwell equations in higher dimensions must be modified in order to have the 1/r-potential as solution for a point charge.

Effective string theory inspired potential and meson masses in higher dimension

“…The formalism can be extended to incorporate such effect in future by adopting higher dimensional Coulomb potential of QCD [25,26,27] suitably. (c)In this work, we have considered all the space dimensions to be large.…”

“…We use (B.13) and (B.14) to numerically calculate < H 1 > given in equation (26) for the case of linear parent.…”

Schrödinger equations on $${\mathbb{R}^3 \times \mathcal{M}}$$ with non-separable potential

“…Some studies on the hydrogen-like atom over higher dimensional Euclidean space include [2][3][4][5][6][7], whereas studies of more complicated atomic structures appear in [8] and [9] for higher dimensional Euclidean space. Nouri describes coherent states for the d-dimensional Coulomb problem in [10], and problems in maintaining regularity of the wave functions for hydrogen atoms defined over higher dimensional Euclidean space are discussed in [11][12][13] whereas Burgbachera, Lämmerzahlb and Maciasc discuss a possible mathematical fix for this problem of regularity in [14].…”

Generalized comparison theorems in quantum mechanics

“…Proof: We suppose that ψ is the eigenfunction corresponding to E N nℓ . We express −∆ in spherical coordinates [1][2][3][4][5][6][7][8][9] and write the radial eigenequation explicitly as…”

“…There is much interest in problems posed in arbitrary dimension N [1][2][3][4][5][6][7][8][9], rather than specifically, say, for N = 1, or N = 3. References [1] and [4] are useful for technical results such as the form of the Laplacian in N -dimensional spherical coordinates; the other papers are concerned with solving problems such as the hydrogen atom [3,7] and the linear, harmonic-oscillator, hydrogen atom, and Morse potentials [8] in higher dimensions than N = 3. The geometrical methods we use in this paper are independent of dimension, which can usually be carried as a free parameter N. We consider examples with Hamiltonians of the form, H = −∆ + v sgn(q)r q , or with sums of such potential terms.…”