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The WKB approximation can be derived using an expansion in powers of $\hbar$. However, that doesn't imply that it can only be used if some quantum number is big. For example, a classic application of the WKB approximation is to alpha decay, which typically occurs from the ground state.

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Always include units Always. No exception. Ever. If your Nobel prize winning professor writes down equations like this without units, you look him in the eye and tell him he's not doing physics. Your analysis suffers from the fatal flaw that you are comparing quantities with different units. $gh/c^2$ is dimensionless. It is the fractional energy change in ...

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Well, first of all, it is important to realize that the integrals (1) and (3) are not merely ordinary double integrals over a single $x$- and a single $p$-variable. Instead they are (Wick-rotated) path integrals containing, heuristically speaking, infinitely many integrations. The path integral derivation of the free particle and the harmonic oscillator ...

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(Not really an answer, but as one should not state such things in comments, I'm putting it here) You commented: "This seems to boil down to the relationship between the phase space and the Hilbert space." That's a deep question. I recommend reading Urs Schreiber's excellent post on how one gets from the phase space to the operators on a Hilbert space in a ...

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