Functional integral (semiclassic formula)

  • Thread starter tpm
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  • #1
tpm
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If to calculate the propagator K(x,x') (vaccuum)for a theory so:

[tex] (i\hbar \frac{\partial}{\partial t}\Psi - H\Psi )K(x,x')=\delta (x-x') [/tex] (1)

we use the functional integral approach:

[tex] K(x,x')=<0|e^{iS[x]/\hbar }|0> [/tex]

my question is, let's suppose we use the semiclassical WKB approach to calculate [tex] K_{WKB} (x,x') [/tex] my question is ¿does the classical propagator satisfies the Schöedinguer equation (1) or as an approximation i'd like to know if the semiclassical propagator satisfies a Hamilton-Jacobi type equation... thanks.
 

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  • #2
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hello
the solution of Hamilton-Jacobi equation is the generators of the canonical transformations (such as the action (the phase) it self) you can see articles: Barut , another : Boujdaa
 
  • #3
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The WKB approximation is a first order expansion in powers of h-bar, so that's how it's related. To zeroth order, you obtain the Hamilton-Jacobi equation, and to first order I believe is where you get the WKB approximation, if I recall correctly.
 

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