Electron-hole overlap integral for a quantum well

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SUMMARY

The discussion focuses on the electron-hole overlap integral Mnn' for quantum wells, specifically addressing cases with infinite and finite barriers. For quantum wells with infinite barriers, Mnn' equals unity when n equals n' and zero otherwise, due to the localization of wave functions. In contrast, for quantum wells with finite barriers, Mnn' is zero when the difference (n-n') is an odd number, reflecting the spatial distribution of the wave functions. These conclusions are derived from the integral M'nn' = ∫φ*en'(z)·φhn(z)dz.

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  • Understanding of quantum mechanics principles, particularly wave functions.
  • Familiarity with quantum well structures and their properties.
  • Knowledge of overlap integrals in quantum physics.
  • Experience with mathematical integration techniques.
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  • Study the properties of quantum wells with infinite barriers in detail.
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  • Learn about the mathematical derivation of overlap integrals in quantum mechanics.
  • Investigate the effects of barrier height on electron and hole localization in quantum wells.
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Hi, I need some help to work out the electron-hole overlap integral Mnn' for a quantum well:
Knowing that M'nn'=\int\varphi*en'(z).\varphihn(z).dz
=> How can I show that Mnn' is unity if n=n' and zero otherwise (in a quantum well with infinite barriers) ?
=> How can I show that Mnn' is zero if (n-n') is an odd number in a quantum well with finite barriers) ?

Thanks a lot
S.
 
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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.
 

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