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Why is it that the WKB approximation produces the correct eigenvalues for the Harmonic Oscillator problem, but the wrong wavefunctions, whereas for the square well, we get correct wavefunctions and the wrong eigenvalues?

I've been trying to dig through the approximations we make in deriving the WKB expressions. The quantization rule that gives the correct eigenvalue for the harmonic oscillator is derived under the condition that near the turning points, the potential can be expanded as a linear function. And V(x) for the SHO is a quadratic function, the "next best" to a linear function.

Can we give a more rigorous reason for this?

Thanks.