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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 particle-in-a-potential model of an atom takes account of the fact that the atomic nucleus is not truly point-like but instead exhibits finite-size 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 non-trivial 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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