Matrix elements for Morse oscillators

Abstract

We have derived exact analytic expressions for the following matrix elements for a Morse oscillator: 〈m‖exp[γ(r−r₀)]‖n〉, 〈m‖(r−r₀)exp[−a(r−r₀)]‖n〉, and 〈m‖(r−r₀)exp[−2a(r−r₀)]‖n〉, where a is the Morse range parameter and γ is an arbitrary constant. We have found that several of the commonly used expressions for various diagonal matrix elements of the Morse oscillator involve considerable roundoff error. Due to near cancellation of terms the result may be many orders of magnitude lower than any of the terms. We obtain simple yet accurate asymptotic approximations for the diagonal elements which avoid these problems. These approximations are obtained for 〈m‖(r−r₀)‖m〉, 〈m‖(r−r₀)²‖m〉, and the cases mentioned above.