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anharmonic oscillation


swaha
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regarding equations of anharmonc oscillation while solving we 1st ignore the anharmonic term say ax square. then we put that solution in the equation and try to solve it again. i dont understand how it reduces the error? please explain.

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regarding equations of anharmonc oscillation while solving we 1st ignore the anharmonic term say ax square. then we put that solution in the equation and try to solve it again. i dont understand how it reduces the error? please explain.

 

We expand the exact solution in series in powers of a small parameter. We consider the anharmonic term as small. So the zeroth-order approximation satisfies the pure harmonic equation. We find it. Then we try to obtain an equation for the first correction. For that we put them together in the equation. We obtain an equation which contains now the known zeroth-order approximation in the anharmonic term. The zeroth-order solution plus the first-order correction is a more accurate solution (smaller error).

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Quantum perturbation theory is in fact easier to deal with, but anyway the frame work of perturbation theory is well established.

 

Any book on quantum theory will talk about perturbation theory and other approximation methods. Classical theory I am less informed about. I do not recall any books on classical mechanics that I have used that deals with this. Any suggestions would be welcomed.

 

What it does require is that the anharmonic term be small compared to the harmonic term. If it is not, then the perturbative expansion is not going to be very reliable (with finite number of terms for sure).


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Yes, I can but I will not. It's you duty.

 

Textbook stuff. swaha can find it out for themselves easy enough.

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