Find the energy eigenvalues and eigenfunctions for the hydrogen atom. The potential

Chapter 13, Problem 22

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Find the energy eigenvalues and eigenfunctions for the hydrogen atom. The potential energy is V (r) = e2/r in Gaussian units, where e is the charge of the electron and r is in spherical coordinates. Since V is a function of r only, you know from that the eigenfunctions are R(r) times the spherical harmonics Y m l (, ), so you only have to find R(r). Substitute V (r) into the R equation in and make the following simplifications: Let x = 2r/, y = rR; show that then r = x/2, R(r) = 2 xy(x), d dr = 2 d dx, d dr r2 dR dr = 2 xy. Let 2 = 2ME/h2 (note that for a bound state, E is negative, so 2 is positive) and = Me2/h2, to get the first equation in 22.26 of Chapter 12. Do this problem to find y(x), and the result that is an integer, say n. [Caution: not the same n as in equation (22.26)]. Hence find the possible values of (these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found proportional to n; let = na, where a is the value of when n = 1, that is, the radius of the first Bohr orbit. Write the solutions R(r) by substituting back y = rR, and x = 2r/(na), and find En from .