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- Degeneracy in one dimension 2 answers
I'm given a particle in a piecewise continous potential $V$ and am supposed to show that normable eigenstates of the Hamilton Operator are not degenerated. My approach was to find two eigenfunctions $\psi_1(x)$ and $\psi_2(x)$ that have the same eigenvalue and the show that they are linear dependent. Plugging into the Schrödinger equation led me to $\psi_1(x) \psi''_2(x) - \psi_2(x) \psi''_1(x) =0$. For linear indepence this equation would only be true for $\psi''_1$ and $\psi''_2$ both being zero, so I thought it's somehow possible to show that they are not zero to show the linear dependence. But I'm stuck so far. The only further assumption I have is that the first differential of the functions is limited for $|x| \to \infty$.