Talk:WKB approximation

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

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

rather than

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

?

See, e.g., http://webpages.ursinus.edu/lriley/courses/p212/lectures/node37.html

That page agrees with the current version of this article. See this equation [1], which says

$k(x)={\sqrt {2m[E-V(x)]}}/\hbar$

and then three equations lower (7.66)

$\psi (x,t)=A_{0}{\sqrt {\frac {k_{0}}{k(x)}}}\exp \left(i\int _{x_{0}}^{x}k(x)dx-\omega t\right)$

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.

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)