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Exactly complete solutions of the pseudoharmonic potential in \(N\)-dimensions. (English) Zbl 1140.81365

Summary: We present analytically the exact solutions of the Schrödinger equation in the \(N\)-dimensional spaces for the pseudoharmonic oscillator potential by means of the ansatz method. The energy eigenvalues of the bound states are easily calculated from this eigenfunction ansatz. The normalized wavefunctions are also obtained. A realization of the ladder operators for the wavefunctions is studied and we deduced that these operators satisfy the commutation relations of the generators of the dynamical group \(SU(1,1)\). Some expectation values for \(\langle r^{- 2}\rangle , \langle r^{2}\rangle , \langle T \rangle, \langle V \rangle, \langle H \rangle, \langle p^{2}\rangle\) and the virial theorem for the pseudoharmonic oscillator potential in an arbitrary number of dimensions are obtained by means of the Hellmann-Feynman theorems. Each solution obtained is dimensions and parameters dependent.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R15 Operator algebra methods applied to problems in quantum theory
81U15 Exactly and quasi-solvable systems arising in quantum theory
33C55 Spherical harmonics
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