Summary: Computer Physics Communications 175 (2006) 315322
A band factorization technique for transition matrix element asymptotics
, I. David Abrahams
School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, England, UK
Received 25 February 2006; received in revised form 9 May 2006; accepted 20 May 2006
Available online 23 June 2006
A new method of evaluating transition matrix elements between wave functions associated with orthogonal polynomials is proposed. The
technique relies on purely algebraic manipulation of the associated recurrence coefficients. The form of the matrix elements is perfectly suited to
very large quantum number calculations by using asymptotic series expansions. In practice, this allows the accurate and fast numerical treatment
of transition matrix elements in the quasi-classical limit. Examples include the matrix elements of xp in the harmonic oscillator basis, and
connections with the Wigner 3j symbols.
© 2006 Elsevier B.V. All rights reserved.
PACS: 03.65.Fd; 03.67.Lx
Keywords: Transition matrix; Quasi-classical approximation; Harmonic oscillator; Orthogonal polynomial
In this paper we offer a new approach to the numerical eval-
uation of transition matrix elements of the form