Is WKB or Perturbation Theory More Applicable for Slowly Varying Potentials?

[Eduard1]

Dear All,

I have recently read about WKB approximation and about perturbation theory.

Both methods are applicable in the range of slowly varying potentials. What I have not understood is which is the range of applicability of one of the method compared with the other one. More precisely: are there cases when one can apply WKB but NOT the perturbation theory ? (or vice versa ?). Also it is not clear to me if one could call WKB a perturbative method or not.

I have tried to get the answer about these 2 questions by looking in some standard QM textbooks (Cohen-Tannoudji) but so far I have not found anything clear. Maybe someone from you with more QM background could give me some hints about the differences btw WKB and perturbative methods.

With all my best wishes,
Ed

 

[Bob_for_short]

Both methods are applicable in the range of slowly varying potentials.

It is no so! WKB is good for "smooth" potentials, the PT is good for "small" potentials. They are quite different. They both are methods of approximating the solutions, but their range of applicability is different. See more examples solved by them to get the difference.

 

[Eduard1]

Dear Bob,

Thanks a lot for the fast answer.

WKB is good for "smooth" potentials, the PT is good for "small" potentials. They are quite different. .

If I do not bother you too much could I ask you which is the difference between "smooth" and "small" potentials. By smooth potential I would understand something like the potential in a metal (i.e. free electron system). But this is also small (I think).

Thanks a lot if you would have some time to clarify me this aspect.

With all my best wishes,
Ed

 

[Bob_for_short]

If I do not bother you too much could I ask you which is the difference between "smooth" and "small" potentials... Ed

WKB takes into account the smooth potential nearly exactly and anyway it does not neglect the potential. Consider a particle trapped in a smooth potential well (a bound state). The WKB solution is very accurate.

PT neglects the potential in the initial approximation, so it should be really small. Then the first correction is searched, the second one, etc. It is just expanding a function in the Taylor series.

I cannot say more. You have to see concrete examples.

 

[Eduard1]

Hi Bob,

Thanks a lot for the answer. It actually helped me to clarify the problem.

Ed

 

[turin]

Actually, WKB is a type of perturbative approach. I suppose that "perturbation theory" is usually used to imply perturbation about a "free" Lagrangian. However, other problems can also be solved by perturbing about an exact solution (e.g. 1/r potential). In fact, the WKB method is basically a perturbative expansion about a Lagrangian that has be gauge-transformed to a free Lagrangian.