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I am reading a book called Sleeping Beauties in Theoretical Physics. In chapter 2 of this book the writer is trying to describe the emergence of classical mechanics from quantum mechanics. I have seen the emergence of classical mechanics from quantum mechanics before from Path Integral. Moreover, I have read this post also. However, I am having problem following this logic. I do not understand his approach (specially the taylor expansion in equation 2.2 and equation 2.8) Addendum: I do know that he is trying to get to Hamilton-Jacobi Equation (2.9) (via Eikonal approximation)

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psimeson
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    I have used Taylor series expansion many times but I do not understand what he is doing in 2.2 and 2.8. The meaning of taylor series expansion there is not that clear to me(which I have already mentioned in thequestion above). I do know that he is trying to get to Hamilton-Jacobi Equation (via Eikonal approximation). – psimeson Dec 10 '16 at 04:56
  • Can you explain a little more what your issue with (2.2) is? How is it any different from other Taylor series you've dealt with? – d_b Mar 15 '19 at 00:33

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  1. Perhaps OP would be more satisfied if we first changed the variable $$\Psi~=~e^{\Phi}\tag{1}$$ in the TDSE so that it becomes a non-linear Riccati equation $$ i\hbar \dot{\Phi}~=~-\frac{\hbar^2}{2m} (\Phi^{\prime\prime}+\Phi^{\prime2}) + V. \tag{2}$$

  2. Now claim that $\Phi$ is a truncated formal Laurent series of the form $$ \Phi~=~\sum_{n=N}^{\infty} \Phi_n \hbar^n , \qquad N\in\ \mathbb{Z}.\tag{3}$$ It is truncated at the term $n=N$ because we expert a correspondence with classical physics. Also note that formal multiplication of truncated Laurent series is well-defined and makes sense. Hence the Riccati eq. (2) becomes an infinite tower of equations.

  3. Next adjust $N=-1$ and rename $\Phi_{-1}=iS_0$, so that the leading equation in eq. (2) is the Hamilton-Jacobi/Eikonal equation.

  4. Finally rename $$\Phi~=~\frac{i}{\hbar} S + \ln R, \tag{4}$$ where $S$ and $R$ are Taylor series in $\hbar$. The variable $\Phi$ may be viewed as an asymptotic expansion in $\hbar$.

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