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 ) = [ x 2 + 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 −4α z − Be −2α z )/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, … , n max ) with the specific property that P ± ( n )Φ n = c ± ( n )Φ n ±1 , where Φ n denotes the n th energy eigenfunction. While P − (0) annihilates the ground‐state Φ 0 , the operator P + ( n max ), instead of annihilating the highest bound‐state Φ , actually knocks it out of the L 2 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 $\hat{P}$ + | n max 〉 = 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 ≡ n max ≡ 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