Since quantum well lasers are made up with heterostructures is there any chance of a photon to be emitted with a different energy of the one related to the well bandgap? I mean the energy of the bandgap of the semiconductor without the well.
Well, the photons emitted with QCL are of lower energy (longer wavelengths).
However, interband transitions in QW lasers can emit high energy photons if wave function overlap of first energy state is less. The radiative transitions probability of higher filled state is larger than lower filled state. In that case, the QW laser could emit higher energy ( shorter wavelength) photons. But its efficiency will be very poor. I think, this is theoretically possible, experimentally it has to be checked.
In a quantum well laser, states are quantized in the conduction and valence bands. Therefore, the transition energy will be greater than the bandgap of the material.