Orthogonality of wave function of finite potential well

[xieyi]

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 !

 

[Avodyne]

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.

 

[Entropia]

The orthogonality of wave functions in a finite potential well is a complex topic that has been extensively studied in quantum mechanics. The reason for the lack of orthogonality between the boundary states in a finite potential well is due to the nature of the potential itself. Unlike an infinite potential well, a finite potential well has a varying potential energy within its boundaries, resulting in a non-constant Hamiltonian. This non-constant Hamiltonian leads to a non-constant eigenvalue, which in turn affects the orthogonality of the wave functions.

To understand this further, we must consider the concept of boundary conditions in quantum mechanics. In an infinite potential well, the boundary conditions are well-defined and the wave function must be zero at the boundaries. However, in a finite potential well, the boundary conditions are not as strict and the wave function can have a non-zero value at the boundaries. This leads to a non-zero overlap between the wave functions of different eigenvalues, resulting in a lack of orthogonality.

Furthermore, the finite potential well also introduces scattering effects, which can further complicate the orthogonality of the wave functions. This is because the wave function of a particle in a finite potential well is a combination of both propagating and evanescent waves, which can interfere with each other and affect the orthogonality of the wave functions.

In summary, the lack of orthogonality between the wave functions of a finite potential well is a result of the varying potential energy and boundary conditions, as well as the introduction of scattering effects. It is a complex phenomenon that requires a deeper understanding of quantum mechanics and the properties of different potential wells.