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# Careers: Physics/Validity of WKB approximation

Question
Hi,

This is not a career related question but i couldn't find anyone else qualified for this question in other sections of this site. I hope you will help.

I have been looking for rigorous mathematical conditions for when the WKB approximation may be applied.

Here is my understanding of the topic.

We start with the most general form that the wavefunction could take, i.e. exp[if(x)/h] ,
Where "i" stands for square root of -1, f(x) is some real function of x, h stands for "h bar",that is the original Planck's constant divide by 2pi.

Any complex function of x can be written this way.

Now we express f(x) as a series in powers of "h bar", i.e.

f(x) = f0 + hf_1 + h^2f_2 + ...

Where again "h" actually stands for "h bar" .

We now put this wave function into schrodinger's one dimensional equation to find various relations.

Now in this entire process, here comes the WKB "approximation" , the approximation being, to neglect h ^2 dependant and all higher terms in the expansion of f(x) i.e. to take f(x) to be
f(x) = f_0 + hf_1

My question: when can we do so ? That is, mathematically, when can we ignore h^2 and higher order terms ?

Thanks

Hi metalrose:

I am more familiar with a slightly different formulation of the WKB approximation.  Basically, the WKB approximation is valid when the derivative of the potential is slow compared to the wavelength of the wavefunction.  In particular the WKB approximation assumes that the potential has a linear spatial term only where it is equal to the energy of the state whose energy (or tunneling probability) is being computed.

Hope this helps,

Carlo

Careers: Physics

Volunteer

#### Carlo Segre

##### Experience

Professor of physics for 30 years at Illinois Institute of Technology. Academic adviser for undergraduates and graduate students. I have served on university promotion and tenure committees, search committees for Deans and Department Chairs. I have also been an Associate Department Chair and an Associate Dean. I have 34 years experience in materials science research and I have been responsible for building and now managing a User facility at the Advanced Photon Source.

Organizations
American Physical Society
Sigma Xi
American Chemical Society
American Associate for the Advancement of Science
International Centre for Diffraction Data (Fellow)
International X-ray Absorption Society

Publications
Nature; Physical Review Letters; Physical Review; Applied Physics Letters; Journal of Physical Chemistry; Journal of Magnetism and Magnetic Materials; Physical Chemistry Chemical Physics; Solid State Communications; Physics Letters; Journal of Low Temperature Physics; Journal of Crystal Growth and Design; Physics Letters; Journal of Applied Physics; Journal of Archaeological Science; Physica C; Corrosion Science; Electrochimica Acta; Journal of Nuclear Materials

Education/Credentials
Ph.D. Physics, 1981 - University of California, San Diego
M.S. Physics, 1977 - University of California, San Diego
B.S. Physics, 1976 - University of illinois, Champaign-Urbana
B.S. Chemistry 1976 - University of illinois, Champaign-Urbana

Awards and Honors
Duchossois Leadership Professor of Physics, IIT Fellow, International Center for Diffraction Data