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Starting with the common variational calculation of the ground-state energy of the helium atom, we calculate the ground-state energy of the lithium atom and then use this result in our determination of the energy levels of the valence electron of lithium. The connected set of calculations is suitable for the typical beginning or intermediate course in quantum mechanics.

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Recently Saleh-Jahromi and Moebs (1998 Eur. J. Phys. 19 355) presented an interesting set of calculations on two-electron (He, 0143-0807/20/1/004/img1, 0143-0807/20/1/004/img2) and three-electron (Li, 0143-0807/20/1/004/img3, 0143-0807/20/1/004/img4) species suitable for undergraduate quantum mechanics courses. This comment offers an extension to these calculations on the three-electron species.

The radial wave functions of inner electron shell and outer electron shell of a Ne atom were obtained by the approximate analytical
method and tested by calculating the ground state energy of the Ne atom. The equivalent volume of electron cloud and the refractive
index of Ne were calculated. The calculated refractive index agrees well with the experimental result. Relationship between
the refractive index and the wave function of Ne was discovered.

A simple Mathematica program for computing the S-state energies and wave functions of two-electron (helium-like) atoms (ions) is presented. The well-known method of projecting the Schrödinger equation onto the finite subspace of basis functions was applied. The basis functions are composed of the exponentials combined with integer powers of the simplest perimetric coordinates. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of results and computation time depend on the basis size. The precise energy values of 7–8 significant figures along with the corresponding wave functions can be computed on a single processor within a few minutes. The resultant wave functions have a simple analytical form consisting of elementary functions, that enables one to calculate the expectation values of arbitrary physical operators without any difficulties.

A simple Mathematica (version 7) code for computing S-state energies and wave functions of two-electron (helium-like) ions is presented. The elegant technique derived from the classical papers of C. L. Pekeris [Phys. Rev., II. Ser. 112, 1649–1658 (1958; Zbl 0082.44604); Phys. Rev. 115, No. 5, 1216–1221 (1959); Phys. Rev. 126, No. 4, 1470–1476 (1962); Phys. Rev. 126, No. 1, 143–145 (1962); Phys. Rev. 127, No. 2, 509–511 (1962)], B. Schiff et al. [Phys. Rev. 140, No. 4A, 1104-1121 (1965)] and Y. Accad and C. L. Pekeris [Phys. Rev. A 4, No. 2, 516–536 (1971)] is applied. The basis functions are composed of the Laguerre functions. The method is based on the perimetric coordinates and specific properties of the Laguerre polynomials. Direct solution of the generalized eigenvalues and eigenvectors problem is used, distinct from the Pekeris works. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of the results and computation times depend on the basis size. The ground state and the lowest triplet state energies can be computed with a precision of 12 and 14 significant figures, respectively. The accuracy of the higher excited states calculations is slightly worse. The resultant wave functions have a simple analytical form, that enables calculation of expectation values for arbitrary physical operators without any difficulties. Only three natural parameters are required in the input. The above Mathematica code is simpler than the earlier version [the authors, Comput. Phys. Commun. 181, No. 1, 206–212 (2010; Zbl 1205.81150)]. At the same time, it is faster and more accurate.

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