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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2015
   • (edited May 26th 2015)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI wanted to add some references on WKB approximation, but found that we have two entries [[semiclassical approximation]] and [[WKB method]]. WKB or semiclassical expansion is one and the same thing: asymptotic expansion of quantum mechanical amplitudes in Planck constant. On the other hand, "WKB method" is often used to limit considerations just to the stationary phase approximation way of doing the expansion, rather than say to the path integral equivalent (the latter anyway used mainly in physics treatments of semiclassical expansion only). 1. Historically WKB or WKBJ (J for Jeffreys) method or approximation has been studied only in one dimension till works of Maslov and others in late 1950s, when the multidimensional analogue has been found. The asymptotics of wave type equations has been studied more generally by Maslov, Hoermander and others as the theory of Fourier differential operators where the stationary phase approximation is the main tool. Mathematically, WKB is precisely the [[stationary phase approximation]] and it has been used much earlier in optics as so called [[geometrical optics]] approximation. One can "historically" limit to just one dimension and just to asymptotics of integral expressions in first order, so in some sense one can limit to some particular case as WKB approximation, but for a modern researcher, WKB and semiclassical method is one and the same thing. I can hardly split the discussion and references to the two entries, so I would rather have them merged into one entry and restrict any mention of the difference in scope to a historical subsection. What do you think about it (Urs, especially). (In fact it makes some sense to rename WKB method entry into 1-dimensional WKB method and to discuss just the old early theory there). 2. I see that the table in [[semiclassical approximation]] says that formal deformation is in all orders while semiclassical just in first (or finite? not clear from the table) order. This is not true, semiclassical expansion is sometimes considered to all orders. But it is an expansion of complex valued functions understood as asymptotic series, and summability issues and analysis of rapidly oscillating functions is in the center of attention. Thus formal means formal, in the sense of formal power series. Semiclassical is asypmtotic expansion, not only formal. Nothing to do with first order ! At [[semiclassical approximation]], I added references on so called exact WKB method, very popular recently, stemming from Voros 1983, where one looks at WKB expansion to _all orders_ and understands it in the sense of [[Borel summability]]. * A. Voros, _The return of the quartic oscillator. The complex WKB method_, Annales de l'institut Henri Poincaré A39:3, 211-338 (1983) [euclid](http://eudml.org/doc/76217) * Alexander Getmanenko, Dmitry Tamarkin, _Microlocal properties of sheaves and complex WKB_, [arxiv/1111.6325](http://arxiv.org/abs/1111.6325) * Kohei Iwaki, Tomoki Nakanishi, _Exact WKB analysis and cluster algebras_, J. Phys. A 47 (2014) 474009 [arxiv/1401.7094](http://arxiv.org/abs/1401.7094); _Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras_, [arXiv:1409.4641](http://arxiv.org/abs/1409.4641)

   I wanted to add some references on WKB approximation, but found that we have two entries semiclassical approximation and WKB method. WKB or semiclassical expansion is one and the same thing: asymptotic expansion of quantum mechanical amplitudes in Planck constant. On the other hand, “WKB method” is often used to limit considerations just to the stationary phase approximation way of doing the expansion, rather than say to the path integral equivalent (the latter anyway used mainly in physics treatments of semiclassical expansion only).

   1. Historically WKB or WKBJ (J for Jeffreys) method or approximation has been studied only in one dimension till works of Maslov and others in late 1950s, when the multidimensional analogue has been found. The asymptotics of wave type equations has been studied more generally by Maslov, Hoermander and others as the theory of Fourier differential operators where the stationary phase approximation is the main tool. Mathematically, WKB is precisely the stationary phase approximation and it has been used much earlier in optics as so called geometrical optics approximation.

   One can “historically” limit to just one dimension and just to asymptotics of integral expressions in first order, so in some sense one can limit to some particular case as WKB approximation, but for a modern researcher, WKB and semiclassical method is one and the same thing. I can hardly split the discussion and references to the two entries, so I would rather have them merged into one entry and restrict any mention of the difference in scope to a historical subsection. What do you think about it (Urs, especially). (In fact it makes some sense to rename WKB method entry into 1-dimensional WKB method and to discuss just the old early theory there).

   1. I see that the table in semiclassical approximation says that formal deformation is in all orders while semiclassical just in first (or finite? not clear from the table) order. This is not true, semiclassical expansion is sometimes considered to all orders. But it is an expansion of complex valued functions understood as asymptotic series, and summability issues and analysis of rapidly oscillating functions is in the center of attention. Thus formal means formal, in the sense of formal power series. Semiclassical is asypmtotic expansion, not only formal. Nothing to do with first order !

   At semiclassical approximation, I added references on so called exact WKB method, very popular recently, stemming from Voros 1983, where one looks at WKB expansion to all orders and understands it in the sense of Borel summability.

   • A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’institut Henri Poincaré A39:3, 211-338 (1983) euclid

   • Alexander Getmanenko, Dmitry Tamarkin, Microlocal properties of sheaves and complex WKB, arxiv/1111.6325

   • Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 26th 2015
   • (edited May 26th 2015)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn order to record the reference by Banerjee that the Hellman-Feynman theorem in [[quantum mechanics]] holds for the first order WKB wave functions, I created the page [[Hellman-Feynman theorem]].

   In order to record the reference by Banerjee that the Hellman-Feynman theorem in quantum mechanics holds for the first order WKB wave functions, I created the page Hellman-Feynman theorem.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 26th 2015
   • (edited May 26th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI seem to remember that we had discussion of this terminological issue before. Since it's about terminology, choose the convention you like best and be sure to add a remark about other possible conventions.

   I seem to remember that we had discussion of this terminological issue before. Since it’s about terminology, choose the convention you like best and be sure to add a remark about other possible conventions.