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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

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