Semi-classical Expansion and the WKB Approximation

One may observe that the Hamilton-Jacobi equation was recovered by assuming the resumed wave function in terms of action  

This result, above of its inverse Legendre transformation form, stands as an implicit equation for the particle s position further identification as 

Nevertheless, the bridge between the classical and quantum regimes may be smoothly accounted from the so-called action semiclassical expansion in powers of h or the eikonal of action  

The replacement of this expansion back into the quantum Hamilton-Jacobi equation provides the successive equations for the various orders of h  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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