Energy and Screening Constants of the Two-Dimensional Helium Atom
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ENERGY AND SCREENING CONSTANTS OF THE TWODIMENSIONAL HELIUM ATOM V. V. Skobelev
UDC 539.12
Following the calculational technique proposed in our previous works, the energy and screening constants of the helium atom with a two-dimensional spatial configuration of its electron distribution have been calculated for the case when one electron is found in the ground state with energy quantum number N 0 , and the second in an excited state with values of the quantum number N 1, 2, ..., 10 . Graphs are presented of the dependence of these quantities on the discrete variable N with elucidation of their characteristic asymptotic behavior. Keywords: screening constants, energy, two-dimensional helium, asymptotics.
INTRODUCTION In [1, 2], the values of the screening constant and the energy E 0 of an ordinary three-dimensional helium atom and also a one-dimensional helium atom were calculated for the excited states of these atoms in which the inner electron is found in the ground state with lowest energy and the outer electron is found in an excited state with the energy quantum number n 2, 3, ..., 9 . In these two papers it was established that both and E fall monotonically with growth of n , and that in the formal limit n , tends to zero and E tends to a constant value that coincides with the energy of a hydrogenlike atom with nuclear charge Z 2 . Since both one-dimensional and twodimensional atoms (specifically, Na [3], see also [4–6]) were obtained by the authors of these works experimentally and the hope exists of obtaining ordinary and lower-dimensional He atoms in similar experiments, it is also of interest to perform an analogous calculation for two-dimensional He atoms.
1. ENERGY AND SCREENING CONSTANTS OF THE TWO-DIMENSIONAL HELIUM ATOM. BASIC EQUATIONS In complete analogy with the calculational technique for the three-dimensional two-electron atom [1, 7] with application of the standard variational method [8, 9] (see also [10]), it is possible to obtain the following expressions for the energy and screening constants of a two-dimensional two-electron atom1 (for Не Z 2 ):
Eat T ( Z ) 2 ,
(1)
1
The applicability of this approximate method in the given situation is not obvious and is also discussed in the last section of our earlier paper [2]. Moscow Polytechnic Institute, Moscow, Russia, e-mail: [email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 152–155, June, 2020. Original article submitted April 8, 2019. 1072
1064-8887/20/6306-1072 2020 Springer Science+Business Media, LLC
2 me c 2 1 1 e2 1 T , , 2 2 c 137 ( N 1/ 2) 2 ( N 1/ 2)
I MM ( K ) I MM ( J ) , T
1
T
(1a)
( N 1/ 2)
2
1 ( N 1/ 2) 2
(1b)
,
(1c)
where
0
0
I MM ( K ) d RN m 2 () d RN m 2 ()
0
0
1 2 d 2 0
1 2 2 2 cos
,
(2a)
I MM ( J ) d RN m () RN m () d RN m () RN m ()
1 2 d 2 0
cos[(m m)]
(2b)
2
2 2 cos
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