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It is well-known that the Hermite functions (not Hermite polynomials!) form a orthonormal basis for a Hilbert space. Therefore, cannot all solutions of the Schrodinger equation even the non-Harmonic oscillator cases be written in terms of Hermite functions?

Thanks.

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    $\begingroup$ In most cases, sure, in the same way we can normally write our solution as a fourier transform or a power series. These are just different choices of basis functions. But generally writing our solutions in terms of Hermite functions is not a helpful thing to do. The coefficients of each basis function will have some typically non-trivial time dependence which there is no particularly simple way to calculate. $\endgroup$ – By Symmetry Aug 25 at 17:44
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    $\begingroup$ @BySymmetry Answers should be posted as answers, not comments. Comments are for asking for clarifications or to suggest improvements to the question. $\endgroup$ – Aaron Stevens Aug 25 at 18:07
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Therefore, cannot all solutions of the Schrodinger equation even the non-Harmonic oscillator cases be written in terms of Hermite functions?

Yes, they can.

However, note that there is a long way between "it is possible" to "it is a useful way to understand the problem". The ver that you can express those solutions in that way doesn't mean it's helpful to do so.

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