INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 37 (2004) 4059–4068 PII: S0305-4470(04)70182-1 Wigner–Eckart theorem in the inductive spaces and applications to optical transitions in nanotubes M Damnjanovi´ c, I Miloševi´ c, T Vukovi´ c and T Marinkovi´ c Faculty of Physics, University of Belgrade, PO Box 368, 11001 Belgrade, Serbia and Montenegro E-mail: yqoq@afrodita.rcub.bg.ac.yu Received 7 October 2003, in final form 16 February 2004 Published 17 March 2004 Online at stacks.iop.org/JPhysA/37/4059 (DOI: 10.1088/0305-4470/37/13/009) Abstract The analytical form of the optical transitions probabilities in carbon nanotubes is found. The derived general form of the Wigner–Eckart theorem for inductive spaces is relevant for any crystal tight-binding model. Within the previously developed modified group projector technique the symmetry based procedure of the matrix elements calculations is obtained. PACS numbers: 78.67.Ch, 61.50.Ah, 02.20, 03.65 1. Introduction The properties of quantum mechanical systems are related (usually through the perturbative approach and transition probabilities) to the matrix elements of the suitably chosen operators between the Hamiltonian eigenstates. It is well known that the symmetry of the system is a powerful tool in such calculations: using symmetry adapted eigenbasis (SAB) and irreducible tensor operators (ITO) one may apply the Wigner–Eckart theorem [1]. Besides the tremendous reduction of the calculations, the Clebsch–Gordan coefficients give the selection rules sublimating all the conservation laws. The one-particle approximation simplifies the structure of the state space of a complex system, retaining only its part induced by the atomic states. Such a framework is particularly convenient for symmetry treatment. In solid state physics the eigenproblem is reduced to the elementary cell, and recently even to the symcell [2], being the contents of the asymmetric unit, i.e. the minimal set of atoms that restore the whole crystal when mapped by the symmetry operations. Less attention has been paid to implement symmetry in the matrix elements calculations for such systems, despite conceptual and efficiency importance. Here we fill in this gap, which is particularly valuable in nanotube physics, due to the high symmetry of nanotubes. At first, we use the modified group projector technique (outlined in section 2) to obtain in section 3 a general expression for the matrix elements of the tensors within the tight-binding model of crystals. It is applied in section 4 to obtain optical transitions 0305-4470/04/134059+10$30.00 © 2004 IOP Publishing Ltd Printed in the UK 4059