# Talk:WKB approximation

WikiProject Physics (Rated B-class, High-importance)

## Untitled

I just removed the long paragraph on WKB's relationship to Feynman diagrams because it was rather unclear (I'm a physics grad student and couldn't tell what it was trying to say) and also seemed to belong more to the article on One-loop Feynman diagrams rather than this page. Laura Scudder 21:48, 23 Feb 2005 (UTC)

Shouldn't the equation be

${\displaystyle \psi (x)\approx {\frac {1}{\sqrt {2m[E-V(x)]}}}\exp \left(\pm {\frac {i}{\hbar }}\int _{-\infty }^{x}dx'2m(E-V(x))\right)}$

rather than

${\displaystyle \psi (x)\approx {\frac {1}{\sqrt {2m[E-V(x)]}}}\exp \left(\pm {\frac {i}{\hbar }}\int _{-\infty }^{x}dx'{\sqrt {2m(E-V(x))}}\right)}$

?

That page agrees with the current version of this article. See this equation [1], which says
${\displaystyle k(x)={\sqrt {2m[E-V(x)]}}/\hbar }$
and then three equations lower (7.66)
${\displaystyle \psi (x,t)=A_{0}{\sqrt {\frac {k_{0}}{k(x)}}}\exp \left(i\int _{x_{0}}^{x}k(x)dx-\omega t\right)}$
so the square root in the exponential is correct, but it should be a 1/4 power in the denominator out front, which I just confirmed in Sakurai. --Laura Scudder | Talk 20:31, 7 May 2005 (UTC)

## One loop diagram

The link to one loop diagrams should be explained. It isn't clear to me at all. The WKB approximation typically yields nonperturbative results. Take e.g. the hydrogen atom in an electrical field. The tunneling rate to the ionized state is proportional to Exp[-constant/|eE|], which is nonperturbative in the coupling (the charge e). To obtain this result from perturbation theory you must perform a resummaton over an infinite number of terms. So, perhaps by 'one loop effect' a summation over all one loop diagram is meant?Count Iblis 22:35, 2 September 2005 (UTC)

## Demonstration

The demonstration given is not conclusive at all.

"The demonstration given is not conclusive at all."

If by that you mean that the derivation is incomplete, then I concur.--Paul 03:22, 26 October 2005 (UTC)

Much nicer now, thanks !

## Missing a bunch of stuff

Ok so long story short, I just spent 30 minutes writing out the WKB method for the tunneling article and, stupid me, I don't think to see if it is already done. However. This article is missing well over half of the solution. I am going to take what I just wrote and fill in all of the missing parts. —Preceding unsigned comment added by C h fleming (talkcontribs) 08:11, 27 October 2005

## Action and classical EM

I'd like to add some discussion on the relationship between the WKB method and action, but I'm not sure to what extent this would clutter the article. I at least alluded to such a connection. Also, if memory serves, there was a similiar method to the WKB method used in classical EM, but I know nothing of this. Can anyone expand on this or at least verify? Threepounds 05:52, 27 November 2005 (UTC)

I have never seen the WKB method used with a Lagrangian, but I have seen other semiclassical approximations. Possibly you mean to talk about the action variable and the early pre-schroedinger equation classical-quantum mechanics that quantized the action variable Int p dq . In the first order of the semiclassical limit, the action variable with classical trajectories is correct. All of these things are related. But that is perhaps more a feature of semiclassical QM in general than the WKB method.
And yes the WKB method did exist before quantum mechanics. It can be used in many PDE's including the Hamilton Jacobi equation. (CHF 13:12, 1 December 2005 (UTC))
Yes, that's essentially what I was trying to get at. I think I agree with your point that that is more an artifact of semiclassical physics then the WKB method itself, so perhaps a bit out of scope. Threepounds 04:31, 2 December 2005 (UTC)

The instanton method in QFT is analogous to the WKB method. See e.g. here. Count Iblis 14:01, 3 December 2005 (UTC)

True. But the instanton method has it's own page. And WKB predates the instanton method. So I think isn't it really for the instanton method's page to mention this and explain it? There seems to be some kind of heirarchy to conserve here? CHF 22:14, 11 December 2005 (UTC)

## Equation numbers?

I'd like to refer to an equation here from another article. The reference would be simpler if there were equation numbers, but perhaps we aren't doing equation numbers in WP? Alison Chaiken 18:35, 21 January 2006 (UTC)

Equation numbers and references/citations have to be put in by hand. In case of references you can use the pair of commands {{ref|label}} and a matching {{note|label}} command, but this doesn't work if you want to refer to a Ref. twice. Perhaps we need to talk to some of the administrators here, because making small edits to articles with a lot of numbered references like this one can be time consuming.Count Iblis 23:23, 21 January 2006 (UTC)
What I ended up doing was putting in this: <div id="mass_in_exponent"></div>. Maybe that's not exactly the right thing to do but it's the best I could figure out from Help:Anchors. I don't quite know what you mean by "{{ref|label}} and a matching {{note|label}} " since the only help pages I find that use "{{" and "}}" are about templates. As far as I can tell, there's no way to search within the "WP:Help" subspace. Alison Chaiken 01:17, 22 January 2006 (UTC)
I used the {{ref|label}} in the article digital physics. You get automatically numbered references that act as links to the citations, but it doesn't work if you refer to the same citation more than once. I didn't previously know about the Anchor command you have used, though!Count Iblis 13:52, 22 January 2006 (UTC)

## Missing Info

Shouldn't this article include a mention of the cases where the WKB approximation gives the exact solution? I don't remember them all off hand, but maybe I'll add them later. Or some else can doesn't matter to me. 24.59.193.0 04:01, 8 March 2006 (UTC)

## Connection formula

It seems that there's an error in the connection formula expressed in terms of Bessel functions. The dimension of the parameter ${\displaystyle U_{1}}$ should be ${\displaystyle [x]^{-3}}$ judging from the simplified differential equation near the turning point ${\displaystyle x_{1}}$. The term (${\displaystyle {\frac {2}{3}}{\sqrt {U_{1}}}(x-x_{1})^{1/3}}$) inside the Bessel functions is supposed to be dimensionless but it's not. Could someone check this point? By the way, this DE can be cast into the form ${\displaystyle y''(x)=x*y(x)}$, the Airy DE. So, the solution can also be expressed by a linear combination of the two Airy functions: ${\displaystyle \psi (x)=C_{1}*AiryAi(U_{1}^{1/3}(x-x_{1}))+C_{2}*AiryBi(U_{1}^{1/3}(x-x_{1}))}$. —Preceding unsigned comment added by 24.59.193.0 (talkcontribs) 04:01, 8 March 2006

## Non-QM approach

While WKB was invented to solve Schrodingers equation, its widely used to solve other wave problems of the form

${\displaystyle {\frac {d^{2}\phi }{dx^{2}}}+k^{2}(x)\phi =0}$

where k varies slowly on the scale of a one wavelength.

This should be mentioned somewhere, and the derivation of the solution, with

${\displaystyle \phi \sim k^{-1/2}\exp({\frac {1}{\epsilon }}\int k(X)dX),\quad X=\epsilon x}$ —Preceding unsigned comment added by Jim McElwaine (talkcontribs) 16:53, 27 April 2006

## Non-QM

I think the page need to be rewritten along the lines of the Non-QM comment at the bottom. The mathematics of the method is very general and obscured by the extra notation h, E V etc. This would make the method easy to understand and to apply to particular cases. The connection to tunneling should be put in a separate page. There is no need to mention Bessel functions at all. The solution is neater in Airy functions and the Airy function page can link to Bessel functions. I am happy to rewrite the page, I'm an applied mathematician at Cambridge, but do not want to make any changes until other people have given their opinions. Jim McElwaine 16:53, 27 April 2006 (UTC)

I've started rewriting it at User:Gareth_Owen/WKB_approximation, but haven't had chance to do more than half. Please complete my job, and we'll present it here when its done. -- GWO
Makes sense. It was developed first independently to solve non-QM problems. — Laura Scudder 18:36, 27 April 2006 (UTC)
I agree, and I have just come across the page as I am teaching the subject later this autumn. Unless anyone objects I will have a go at implementing the above suggestion when I get around to itBilllion (talk) 08:50, 5 September 2008 (UTC)
As far as I can see, the above already has been implemented, by 140.247.248.31 in November 2007. -- Crowsnest (talk) 09:31, 5 September 2008 (UTC)
The example section is appropriate, but the application to Schrodinger equation seems to me inappropriate. Perhaps it should be moved to a wikibook? Also, contrary to the introduction which states WKB is applicable to linear PDEs, the method section implies the method is applicable to linear homogeneous ODEs. All this, and it seems to me at after an hour's pondering, that the method has a good chance of helping one solve any DE where the highest order derivative dominates the lower order derivatives... I wish this idea had been in the first line of the article. 169.237.31.150 (talk) 02:49, 28 January 2012 (UTC)

## wrong equation ?!

Psi=exp(phi) and Psi'=A+iB, but then the following statement below is not true, cause the DERIVATIVE of Psi equals A+iB and not Psi directly:

The amplitude of the wavefunction is then exp[A(x)] while the phase is B(x).

Better remove it completely?62.218.164.30 17:54, 17 November 2006 (UTC)

## Wrong equation!

The derivative of Psi equals (A'+iB')Psi! The equations for the functions A and B are wrong. Plugging in the Ansatz

${\displaystyle \Psi (x)=e^{\Phi (x)}}$
${\displaystyle \Phi '(x)=A(x)+iB(x)}$

into the Schrödinger equation leads to

${\displaystyle A''+A'^{2}-B'^{2}={\frac {2m}{\hbar ^{2}}}(V-E)}$
${\displaystyle B''+2A'B'=0}$

. —The preceding unsigned comment was added by Matrix1329 (talkcontribs) 11:06, 6 January 2007 (UTC).

Matrix1329 11:09, 6 January 2007 (UTC)

You seem to working with ${\displaystyle \phi (x)=A(x)+iB(x)}$ rather than the stated ${\displaystyle \phi '(x)=A(x)+iB(x)}$. That's why you have second derivatives of A and B running around. — Laura Scudder 14:36, 1 December 2007 (UTC)

## Lacking basics

This page provides a lot of maths, suited more to a physics textbook and not an encyclopedia. I'm not saying anything should be removed, but simpler descriptions of the concepts should ideally be provided to those not of a physics-related background. LaudanumCoda 14:50, 10 May 2007 (UTC)

This is a topic which one would meet in the final year of a physics undergrad degree. Might be difficult to clearly give a good idea. 82.16.99.131 (talk) 10:12, 4 November 2008 (UTC)

## Higher Dimension?

I think it should be added the WKB formulation for more than one dimension when SE ain't separable , for example whenever you have the potential ${\displaystyle V(x,y,z)}$ in 3-D of the form V=xyz so you need to evaluate the solution using WKB approach, i think Einstein and others estudied this problem but not sure --85.85.100.144 11:31, 26 June 2007 (UTC)

I added some details of the WKB method and an example from Bender and Orszag. I can be contacted at [redacted] (at) gmail if you think the article still lacks clarity. (Or if you want to thank me for enabling you to do your Applied Math 201 problem set. I'm talking to you, Harvard students.)

--140.247.248.31 (talk) 17:53, 26 November 2007 (UTC)

## History

In: Olver, Frank J. W. (1974). Asymptotics and Special Functions. Academic Press. ISBN 0-12-525850-X., p.228, there is an account on the history of the WKBJ method. References therein are:

• Francesco Carlini (1817). "Richerche sulla convergenza della serie che serva aal soluzione del problema di Keplero". Milano.
• Joseph Liouville (1837). "Sur le développement des foncttions et séries ...". Journal de Mathématiques Pures et Appliquées 1:16-35.
• George Green (1837). "On the motion of waves in a variable canal of small depth and width". Transactions of the Cambridge Philosophical Society 6:457-462.
• Lord Rayleigh (1912). "On the propagation of waves through a stratified medium, with special reference to the question of reflection". Proceedings of the royal Society London, Series A 86:207-226.
• Richard Gans (1915). "Fortplantzung des Lichts durch ein inhomogenes Medium". Annalen der Physik [4] 47:709-736.

Liouville (1837) and Green (1837) are credited by Olver as the ones who developed the method. Further, in the same historical account by Olver, he cites Jeffreys (1953), who refers to Gans (1915) and (to a lesser extend) Lord Rayleigh (1912) as earlier accounts of the method. -- Kraaiennest (talk) 02:10, 25 January 2008 (UTC)

## Abbreviation alternatives

The abbreviation JWKB has been introduced, to replace WKBJ. I did a Google Scholar search to get an indication on the use of the various forms:

• "WKB approximation": 15500 hits
• "JWKB approximation": 1030
• "WKBJ approximation" : 834
• "WBK approximation" : 57
• "BWK approximation" : 33
• "BWKJ approximation" : none, but Google books finds one
• "Wentzel-Kramers-Brillouin approximation" : 529
• "Jeffreys-Wentzel-Kramers-Brillouin approximation" : 12
• "Wentzel-Kramers-Brillouin-Jeffreys approximation" : 7

WKB is by far the most used expression for the method, while JWKB and WKBJ are both quite often used. To my opinion, this should be reflected somehow in the article. -- Crowsnest (talk) 08:43, 18 March 2009 (UTC)