Generalized hypervirial theorem for the D dimensional single-particle system and its applications

By using the Hamiltonian identity, we present a generalized hypervirial theorem for the D dimensional single-particle system with arbitrary potential. It is shown that this generalized hypervirial theorem is powerful in deriving the Blanchard's and Kramers' recurrence relations among the matrix elements. We apply those recurrence relations to some physical systems, exactly solvable and unsolvable, such as the pseudoharmonic oscillator, the Morse, the modified Pöschl-Teller, the Lennard-Jones, the Buckingham and the Yukawa potentials. The Blanchard's and Kramers' recurrence relations in two dimensions are also briefly mentioned.