Consider the infinite spherical well of radius a. in lecture we found that the solution to the radial equation was u(r) = ar je(kr). in this problem we will consider the solutions when l = 1. (a) extra part (not for credit) show that j1(x) = sinc cosc c22 2 - using the definition j(x) = (-2)* () (sinu). (b) extra part (not for credit) find and sketch the effective potential veff. e(r) for l = 1 and explicitly check that ur) = arji(kr) solves the one-dimensional tise for this potential. let me be the n-th zeros or roots of the l-th spherical bessel functions so that bu is the first zero of j1(2), b21 is the second zero of ji(2), etc. just as we did with the one-dimensional finite square well, we can use graphical methods to find these roots and thus the allowed energies. (c) show that j1(x) = 0 implies x = tanx. plot x and tanx on the same graph and locate the points of intersection (the x values of these intersection points gives us our bessel function zeros bni). note that x = 0) and thus e= 0 is a solution. even though x = 0 solves x = tan z we don't use it to give us a physically allowed energy. why not? [note: your graph doesn't have to be too fancy. you can just make a sketch, you don't need to use a grapher or print anything out. nor do you do need to numerically solve for the roots in this part.) (d) argue from your graphical solution that, for large n, the energies satisfy h272 eni ~ m2 (n + ). (1) (e) using whatever method you'd like, find the energies en,1 for n=1 and n = 5. express these energy as a number times h272 /2ma? . f) going back to the full 3d problem, what is the degeneracy of the n = 1, 1 = 1 state? write out the full wave function 0110(r, 0,0) explicitly (the indices on y mean n = 1, 1 = 1, and m = 0).2 you do not have to solve for the normalization constant.