The Quasi Classical Approximation(Landau Lifshitz)

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So I was reading the Landau Lifgarbagez book on non-relativistic quantum mechanics and ran into this quasi-classical approximation they use at various points in the book.They have argued with an analogy that in the classical limit, the phase of the wave function will be proportional to the classical action S, like in optics.Can somebody give a more digestible insight than that?

And then in the quasi-classical chapter,they argue that the system is almost classical and the action integral(time-independent part)can be written as a power series of h.How do you justify that?
 

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