The evaluation of the matrix element for interband optical transitions in quantum wells using envelope functions

The expression for the matrix element of position for an interband transition in a wide deep quantum well is derived using the envelope function expansion. It is found that the matrix element is not determined by the intra-atomic-like matrix element evaluated using band-edge Bloch functions, the result one might be led to expect either on intuitive grounds or as the result of a supposedly approximate evaluation in which only the dominant term in the envelope function expansion of each wave function is used. To obtain the correct expression, terms in the envelope function expansion that become vanishingly small in the limit of wide wells must be retained. Usually the interband matrix element is comparable with that for allowed intersubband transitions. Some implications for the DC Stark effect for quantum wells and non-linear refraction in quantum dot structures are mentioned.