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Unformatted text preview: Physics 6210/Spring 2007/Lecture 31 Lecture 31 Relevant sections in text: 5.1, 5.2 Example: finite size of the atomic nucleus One improvement on the simple particleinapotential model of an atom takes account of the fact that the atomic nucleus is not truly pointlike but instead exhibits finitesize structure in both its mass and charge distributions. It is possible to use perturbation theory to get a quick look into the effect on atomic spectra of this feature. Of course, this effect is one of a myriad of small corrections that need to be added to the atomic model. Let us model the nucleus as a (very massive) uniform ball of total charge Ze with radius r . As a nice exercise in elementary electrostatics, you can check that the potential energy V ( r ) = e for such a charge distribution takes the form V ( r ) =  Ze 2 r for r r Ze 2 2 r r r 2 3 , for r r . I do not know if a closed form solution to the eigenvalue problem for the Hamiltonian H = P 2 2 m + V is known, but I doubt that such a solution exists. We well treat the potential energy due to the finite size r of the nucleus as a perturbation of the usual Coulomb potential. To this end we write V ( r ) = V ( r ) + B ( r ) , where V ( r ) = Ze 2 r , for r > , and B ( r ) = Ze 2 2 r r r 2 3 + 2 r r for 0 r r , for r r . The idea is then that, since the unperturbed energy eigenfunctions are nontrivial over a range corresponding to the Bohr radius a (for the given Z ), as long as r << a we expect that the effect of the perturbation B will be small. We will make this more precise in a moment. 1 Physics 6210/Spring 2007/Lecture 31 Recall that the energy eigenstates  n,l,m i have position wave functions given by h r  n,l,m i nlm ( r,, ) = R nl ( r ) Y lm ( , ) , where the Y lm are the usual spherical harmonics and R nl ( r ) = " 2 na 3 ( n l 1)!...
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 Spring '07
 M
 Physics, mechanics

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