Abstract
There has been a long history of attempts to calculate the matrix elements and the recurrence relations among them for some important wavefunctions such as the Coulomb-like potential, harmonic oscillator, Kratzer oscillator and others because of their wide applications. It should be pointed out that almost all contributions appearing in the literature have been made in three dimensions. In this Chapter, we first present a generalized second hypervirial formula in dimensions D and then obtain the general Blanchard’s and Kramers’ recurrence relations. After that, we shall apply them to obtain the corresponding Blanchard’s and Kramers’ recurrence relations for those certain central potentials such as the Coulomb-like potentials, the harmonic oscillator, Kratzer oscillator and the Morse potential.
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Notes
- 1.
It is worth addressing that the “Coulomb-like” potential in almost all contributions mentioned above and others [326, 327] has the form 1/r. Even though the real Coulomb-like potential in two dimensions is taken as a logarithmic form \(\ln r\), its exact solutions have not been obtained except for the approximate solutions [328].
References
Flügge, S.: Practical Quantum Mechanics. Springer, Berlin (1971)
Louck, J.D., Shaffer, W.H.: Generalized orbital angular momentum and the n-fold degenerate quantum-mechanical oscillator: Part I. The twofold degenerate oscillator. J. Mol. Spectrosc. 4, 285–297 (1960)
Louck, J.D.: Generalized orbital angular momentum and the n-fold degenerate quantum-mechanical oscillator: Part II. The n-fold degenerate oscillator. J. Mol. Spectrosc. 4, 298–333 (1960)
Louck, J.D.: Generalized orbital angular momentum and the n-fold degenerate quantum-mechanical oscillator: Part III. Radial integrals. J. Mol. Spectrosc. 4, 334–341 (1960)
Chatterjee, A.: Large-N expansions in quantum mechanics. Phys. Rep. 186, 249–370 (1990)
Waller, I.: Der Starkeffekt zweiter Ordnung bei Wasserstoff und die Rydbergkorrektion der Spektra von He und Li+. Z. Phys. 38, 635–646 (1926)
Van Vleck, J.H.: A new method of calculating the mean value of 1/r s for Keplerian systems in quantum mechanics. Proc. R. Soc. Lond. A 143, 679–681 (1934)
Pasternack, S.: On the mean value of r S for Keplerian systems. Proc. Natl. Acad. Sci. USA 23, 91–94 (1937). Erratum in Proc. Natl. Acad. Sci. USA 23, 250
Pasternack, S., Sternheimer, R.M.: An orthogonality property of hydrogenlike radial functions. J. Math. Phys. 3, 1280 (1962)
Armstrong Jr., L.: Group properties of hydrogenic radial functions. Phys. Rev. A 3, 1546–1550 (1971)
Ding, Y.B.: On the Schrödinger radial ladder operator. J. Phys. A, Math. Gen. 20, 6293 (1987)
Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Academic Press, New York (1957)
Blanchard, P.: A new recurrence relation for hydrogenic radial matrix elements. J. Phys. B, At. Mol. Phys. 7, 993 (1974)
Bockasten, K.: Mean values of powers of the radius for hydrogenic electron orbits. Phys. Rev. A 9, 1087–1089 (1974)
Hughes, D.E.: Recurrence relations for radial matrix elements obtained from hypervirial relations. J. Phys. B, At. Mol. Phys. 10, 3167 (1977)
Drake, G.W.F., Swainson, R.A.: Expectation values of r P for arbitrary hydrogenic states. Phys. Rev. A 42, 1123 (1990)
Drake, G.W.F., Swainson, R.A.: Erratum: Expectation values of r P for arbitrary hydrogenic states. Phys. Rev. A 43, 6432–6432 (1991)
Swainson, R.A., Drake, G.W.F.: An alternative proof of some relations between hydrogenic matrix elements. J. Phys. B, At. Mol. Opt. Phys. 23, 1079 (1990)
Ojha, P.C., Crothers, D.S.: On a simple relation between hydrogenic radial matrix elements. J. Phys. B, At. Mol. Phys. 17, 4797 (1984)
Shertzer, J.: Evaluation of matrix elements 〈n,l‖r β‖n,l′〉 for arbitrary β. Phys. Rev. A 44, 2832–2835 (1991)
Moreno, B., Piñeiro, A.L., Tipping, R.H.: Algebraic solution for the hydrogenic radial Schrödinger equation: matrix elements for arbitrary powers of several r-dependent operators. J. Phys. A, Math. Gen. 24, 385 (1991)
Qiang, W.C., Dong, S.H.: An alternative approach to calculating the mean values of \(\bar{r^{k}}\) for hydrogen-like atoms. Phys. Scr. 70, 276–279 (2004)
Kramers, H.A.: Quantum Mechanics. North-Holland, Amsterdam (1957). Sect. 59
Sánchea, M.L., Moreno, B., López Piñeiro, A.: Matrix-element calculations for hydrogenlike atoms. Phys. Rev. A 46, 6908–6913 (1992)
Morales, J., Peña, J.J., Portillo, P., Ovando, G., Gaftoi, V.: Generalization of the Blanchard’s rule. Int. J. Quant. Chem. 65, 205–211 (1997)
López-Bonilla, J.L., Morales, J., Rosales, M.A.: Hypervirial theorem and matrix elements for the Coulomb potential. Int. J. Quant. Chem. 53(1), 3–7 (1995)
Núñez-Yépez, H.N., López-Bonilla, J.L., Salas-Brito, A.L.: Generalized hypervirial and recurrence relations for hydrogenic matrix elements. J. Phys. B, At. Mol. Opt. Phys. 28, L525 (1995)
Martínez-y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: Relativistically extended Blanchard recurrence relation for hydrogenic matrix elements. J. Phys. B, At. Mol. Opt. Phys. 34, 1261 (2001)
Martínez-y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: A useful form of the recurrence relation between relativistic atomic matrix elements of radial powers. J. Phys. B, At. Mol. Opt. Phys. 35, L71 (2002)
Dong, S.H., Chen, C.Y., Lozada-Cassou, M.: Some recurrence relations among the radial matrix elements for the relativistic hydrogenic atoms. Phys. Lett. A 333, 193–203 (2004)
Basida, A., Zúñiga, J., Alacid, A., Requena, A., Hidalgo, A.: Two-center matrix elements for Kratzer oscillators. J. Chem. Phys. 93, 3408 (1990)
Morales, J.: Generalized recurrence relation for the calculation of two-center matrix elements. Phys. Rev. A 36, 4101–4103 (1987)
Alves, N.A., Drigo Filho, E.: The factorisation method and supersymmetry. J. Phys. A, Math. Gen. 21, 3215 (1988)
Yang, X.L., Guo, S.H., Chan, F.T., Wong, K.W., Ching, W.Y.: Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory. Phys. Rev. A 43, 1186–1196 (1991)
Guo, S.H., Yang, X.L., Chan, F.T., Wong, K.W., Ching, W.Y.: Analytic solution of a two-dimensional hydrogen atom. II. Relativistic theory. Phys. Rev. A 43, 1197–1205 (1991)
Müller-Kirsten, H.J.W., Bose, S.K.: Solution of the wave equation for the logarithmic potential with application to particle spectroscopy. J. Math. Phys. 20, 2471 (1979)
Huffaker, J.N., Dwivedi, P.H.: Diatomic molecules as perturbed Morse oscillators. IV. Franck-Condon factors for very high J. J. Chem. Phys. 68, 1303 (1978)
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Dong, SH. (2011). Generalized Hypervirial Theorem. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_10
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