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Hydrogen equation in spaces of arbitrary dimensions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Conf. Ser. 635 042001 (http://iopscience.iop.org/1742-6596/635/4/042001) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 154.127.58.215 This content was downloaded on 17/04/2017 at 14:07 Please note that terms and conditions apply.
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XXIX International Conference on Photonic, Electronic, and Atomic Collisions (ICPEAC2015) IOP Publishing Journal of Physics: Conference Series 635 (2015) 042001 doi:10.1088/1742-6596/635/4/042001
Hydrogen equation in spaces of arbitrary dimensions M. Ya. Amusia*† 1 *
†
The Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel A. F. Ioffe Physical-Technical Institute, St. Petersburg 194021, Russian Federation
Synopsis We note that presenting Hydrogen atom Schrodinger equation in the case of arbitrary dimensions require simultaneous modification of the Coulomb potential that only in three dimensions has the form Z / r . This was not done in a number of relatively recent papers (see [1] and references therein). Therefore, some results obtained in [1] seem to be doubtful. Several required considerations in the area are mentioned.
1. The aim of the article [1] is to show how to treat using old Bohr’s approach not only simple atoms but molecules. An important point in this development is reconciliation of quantum mechanics with Bohr’s ideas. It is stated in [1] that in the infinite dimension quantum mechanics “morphs into classical mechanics”. However, the infinite-dimension system is not a clarifying model to describe physical or chemical objects. Far from being convincing are statements like “Hence the large-D limit, where 1/Dĺ0 is closer to the real world (1/D=1/3) than is the oftenused D=1 regime. Indeed, results obtained at large D usually resemble those for D=3”. I must confess, it sounds too light weighted to be convincing. 2. But not the large-D limit per se is doubtful. Of great concern is the assumption that the radial part of the D-dimensional Schrödinger equation in Hartree units looks as follows: 1 ∂ 2 [l + ( D − 3) / 2][l + ( D − 1) / 2] Z ½ + − ¾ φ = Eφ , (1) ®− 2 2r 2 r¿ ¯ 2 ∂r
where Z is the nuclear charge and l is the angular momentum. Obviously, this equation has no sense for D = 1 , since a finite angular momentum requires infinite speed of a rotating particle. 3. The equation (1) implies independence of the Coulomb potential upon D. However, to obtain the Coulomb law for a two-dimensional space, one has to consider Maxwell equations in a twodimensional world. This equation for the considered case looks as § ∂2 ∂ 2 · (2) (2) − + ϕ (r ) = Zδ (2) (r ) , ¨ 2 © ∂x
¸ ∂y 2 ¹
where δ (2) ( r ) is the two-dimensional deltafunction and r 2 = x 2 + y 2 . This leads to
ϕ (2) (r ) = −2 Z ln(r / r0 )
¦
i =1
∂xi2
Here r 2 = ¦ i =1 xi2 . i=D
This derivation would also permit to check whether indeed at D → ∞ one arrives to a simple classical picture that I doubt. Quite interesting would be an attempt to model experimentally the transition from Z / r to (3) by compressing an electron-proton-like system using analog of a flat conducting mirrors. Perhaps that could be achieved in cavities of rectangular shape. 7. There exists another case, when the Coulomblike potential is of interest. I mean Newton’s gravitation law. Its analogue for two-dimension world is definitely of at least theoretical interest. To do this, one has to use the equation of Einstein’s general relativity and to apply them for a massive body in the field of point-like source of gravitational field. References
(3)
instead of Z / r r-dependence in the three dimensional case. Here r0 is a cutoff length that depends upon the problem at hand. 1
4. If one takes into account the D-dependence of the Coulomb potential, derivations in [1] became meaningless. Indeed, by using an unrealistic equation (1), how one can believe that it illuminates quite realistic and physical postulates of Bohr?! It is known since long ago a number of papers that consider two-dimensional Hydrogen atom (see e.g. [2]) that use for this or that reason the incorrect for the two-dimensional case potential Z / r instead of the correct one (3). It would be desirable to perform calculations with (3), instead of Z / r . 5. It would be of interest to find for completeness the shape of a “Coulomb” potential in the case of any arbitrary dimension D>3. This could be achieved by solving the equation 2 i =D ∂ (5) ϕ (2) (r ) = Z δ (2) (r ) . −
[1] A. Svidzinsky, M. Scully, and D. Herschbach 2014, Physics Today, 1, 33. [2] ]. G. Q. Hassoun 1981 Am. J. Physics, 49 (2), 143
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