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Comments to the question (v3): Yes, given a Hamiltonian $H$, the semiclassical WKB method can give a qualitative but not a quantitative prediction for the ground state energy $E_0$. The WKB prediction (incl. the metaplectic correction/Maslov index) for the harmonic oscillator (HO) happens to be exact due to a hidden supersymmetry, cf. this Phys.SE post. ...

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