The Coulomb problem in two dimensions: One possible view of the Coulomb problem

Question

The Coulomb problem in two dimensions: One possible view of the Coulomb problem and the hydrogen atom is that Bohr?s picture of atomic orbits is partially correct, with the electrons moving in two-dimensional orbits around a fixed center. In this case, in polar coordinates (r psi) the Schrodinger equation is: Here, M is the mass of the electron. After performing the separation of variables, the wave function psi (r, phi ) is given by psi(r,phi)= R(r) phi(Phi). The angular wave function phi (phi) =e^ imp phi where m is an integer that can be 0,+ or - 1, + or - 2, ... etc. Part 1. The ground-state wave function corresponds to the situation with m=0, and the radial wave function is given by: R(r) = Ae ^-Beta r ,where A is a normalization constant and Beta is a constant to be determined. (a) Using an approach similar to that illustrated by Krane to solve the id harmonic oscillator problem (page 156), show that with m=0, this radial function satisfies Schrodinger?s equation with a proper choice of Beta and E. Compare the value of E to the known value of the energy of the ground state of the hydrogen atom. Do you get the right answer? (That is, does the solution to this problem give the correct value of the hydrogen ground-state energy?) (b) Since m tells us the angular momentum L=mh, it seems that the correct association with the Bohr model of the hydrogen atom should have m=+ or -1, rather than 0. In this case, the radial wave function is given by R(r) = Are^ - beta r. Show that this radial function also satisfies the Schrodinger equation, and determine the correct values of Beta and E. How does the energy here compare to the known ground-state energy of the hydrogen atom, and to the value of the energy determined in part (a)?

Explanation / Answer

The length scale L has to be present in the denominator for dimensional reasons

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