Explicit Relations of Physical Potentials Through Generalized Hypervirial and Kramers' Recurrence Relations

Based on a Hamiltonian identity, we study one-dimensional generalized hypervirial theorem, Blanchard-like (non-diagonal case) and Kramers' (diagonal case) recurrence relations for arbitrary x^κ which is independent of the central potential V(x). Some significant results in diagonal case are obtained for special κ in x^κ (κ ≥ 2). In particular, we find the orthogonal relation 〈n₁|n₂〉 = δ_n1n2 (κ = 0), 〈n₁|V′(x)|n₂〉 = (E_n1 - E_n2)²〈n₁|x| n₂〉 (κ = 1), E_n = 〈n|V′ (x)x/2|n〉 + 〈n|V(x)|n〉 (κ = 2) and -4E_n〈n|x|n〉 + 〈n|V′(x)x²|n〉 + 4〈n|V(x)x|n〉 = 0 (κ = 3). The latter two formulas can be used directly to calculate the energy levels. We present useful explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.