Home › Forums › Hyperfine course › a nucleus with a general shape ? › Perturbation Theory and Multipole Expansions
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February 28, 2024 at 12:15 am #5164CmrossiParticipant
If you try to study a general shaped nucleus without making a multipole expansion first, you can run into complications with systems that can be unsolvable. Thus, you need to make a multipole expansion in order to take into account the part of the multipole that contributes the most to what you’re calculating. This breaks down the calculations into more basic calculations (which are still complex, but much more easily calculated than the original).
A multipole expansion is valid when the higher order terms decrease in the overall numerical size (i.e. the expansion converges quickly enough). This makes sure that the multipole expansion converges to the true value of the original problem if you take an infinite number of higher order terms. If the expansion did not converge quickly enough, you would need to take more and more higher order terms into account since they would be too large to ignore. Another issue is that if the expansion does not converge quickly enough, then your values may become infinite, so care needs to be taken when using the multipole expansion. When the expansion does converge quickly enough, then truncating it at first order or second order is good enough of an approximation for reality as all the higher order terms decrease in size very quickly.
Perturbation theory is valid only if when the quantum solutions to original, unperturbed system are already known. That is, if you treat the perturbation as an extra potential added on to the Hamiltonian, then perturbation theory is valid if you know the solution to the original Hamiltonian.
Both the requirements to truncate the multipole expansion and use perturbation theory simultaneously can be realized if the size of the deformation of the nucleus is smaller than the normal distance between the nucleus and the electron.
I could be wrong on all of these, I honestly don’t know if my reasoning is correct or not.

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