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The semiclassical, or WKB, approximation is one that is far more natural in the path integral formalism than it is when derived from the Schrodinger equation directly.

Furthermore, the connection formulae that relate solutions to the left and right of a turning point in this approximation are awkward, and their derivation in terms of Airy functions leaves quite a bit to be desired.

Thus the thesis of my question: is there a simple way to derive the connection formulae directly from the path integral formalism? In particular, I'd be very interested to see the origin of the $e^{\pm i\pi/4}$ terms come from something a little more physically satisfying than "it just comes from the asymptotic forms of the Airy functions!"

Bob Knighton
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  • I don't know the full story here, but the keywords you're looking for are "Morse index" and "van Vleck formula", which talk about the extra phases you get for paths directly from a semiclassical approximation of the path integral. For example, see the paper here. It's probably also somewhere in the 20 appendices of Arnold. – knzhou Mar 08 '19 at 12:40
  • Regarding path integrals in relation to the pi/4 angle you refer to (but not explicitly linking to WKB) see the details section of this answer: https://physics.stackexchange.com/a/384919/83405 – Wakabaloola Mar 09 '19 at 11:33
  • For the standard (non-path-integral) derivation of the WKB connection formulae, see e.g. this Phys.SE post. – Qmechanic Mar 09 '19 at 16:43
  • You could also have a look to the "canonical" book of Landau and Lifchitz, "Quantum mechanics" where the pi/4 dephasing is obtained without Airy function, but at the expense of an integral on complex coordinate. But after all the Airy function is characteristic of any kind of caustics surfaces, reduced to the turning points in 1D. – Jhor Mar 13 '19 at 09:30

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The book by Schulman on path integrals goes into this problem with quite a bit of detail (multiple chapters on it).

Quickly glancing through Ch. 14 as a refresher, it appears that the WKB method is merely the stationary-phase approximation to the path integral (treating $1/\hbar$ as a large-$N$ expansion); except one tries to solve the resulting equations in equivalent orders of $1/\hbar$. I don't have time for a more thorough analysis, and this is probably a bit late anyway, but I felt like answering anyway!

Thomas Fritsch
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Slenderman
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