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].
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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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