Ladder operators for the Kratzer oscillator and the Morse potential

Abstract

Abstract Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x2 + β(β − 1)x−2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae−4αz − Be−2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P±(n) (n = 0, 1, 2, 3, … , nmax) with the specific property that P±(n)Φn = c±(n)Φn±1, where Φn denotes the nth energy eigenfunction. While P−(0) annihilates the ground‐state Φ0, the operator P+(nmax), instead of annihilating the highest bound‐state Φ , actually knocks it out of the L2 space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators $\hat{P}$ ± with proper end‐of‐spectrum behavior (i.e., $\hat{P}$ −|0〉 = 0 and $\hat{P}$ +|nmax〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators $\hat{P}$ +, $\hat{P}$ −, and $\hat{P}$ 0 (where $\hat{P}$ 0 ≡ (1/2)[$\hat{P}$ +, $\hat{P}$ −]) form a su(2) algebra only if we restrict them to the (N − 1)‐dimensional subspace spanned by the lowest (N − 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ nmax ≡ integral part of (1/2)(B/(2α$\sqrt{A}$ ) − 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N − 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006