Electron-hole overlap integral for a quantum well

[skyboarder2]

Hi, I need some help to work out the electron-hole overlap integral M_nn' for a quantum well:
Knowing that M'_nn'=$$\int$$$$\varphi$$^*_en'(z).$$\varphi$$_hn(z).dz
=> How can I show that M_nn' is unity if n=n' and zero otherwise (in a quantum well with infinite barriers) ?
=> How can I show that M_nn' is zero if (n-n') is an odd number in a quantum well with finite barriers) ?

Thanks a lot
S.

 

[azntoon]

For a quantum well with infinite barriers, the electron wave function (en) and hole wave function (e-h) are localized in the same region of space. This means that the integral M'nn' is equal to the overlap of the two wave functions, which is unity if n=n' and zero otherwise. For a quantum well with finite barriers, the electron and hole wave functions can be spread into different regions of space. In this case, the integral M'nn' is equal to the overlap of the two wave functions, which is zero if (n-n') is an odd number.