Orthogonality of wave function of finite potential well

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The discussion centers on the orthogonality of wave functions in finite potential wells. While wave functions in infinite potential wells form a complete orthogonal basis, the user xieyi observed that the wave functions for finite potential wells appear non-orthogonal. However, it is established that the eigenfunctions for finite potential wells are indeed orthogonal, despite the complexity involved in proving this through direct calculation.

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xieyi
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Hello,

As we know, the wave function of infinite potential wells form a complete orthogonal base. I have tried now to solve out the wave function for finite potential well, checking the orthogonality, I found that they are no longer orthogonal to each other (I mean the wave function of those boundary states). This is not that understandable to me, since the Hamiltonian is Hermite and the resulting wave function from different eigenvalue (non-degeneracy) should be "always" orthogonal to each other. Could anyone give explanation ?

Thank you !
 
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Welcome to the forum, xieyi.

The eigenfunctions for the finite well are orthogonal. To show this by direct calculation is pretty messy, but it can be done.
 

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