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(Solved): Consider the long-range Coulomb potential in the hydrogen atom shielded for small distances. The shi ...
Consider the long-range Coulomb potential in the hydrogen atom shielded for small distances. The shielding should be modeled by a quadratic potential, as you know it from the harmonic oscillator
(a) Determine the parameters ? and V_0 in such a way that the two potentials at the point r_0 merge into one another in a continuously differentiable manner and plot the potential together with the unshielded Coulomb potential for r_0 = a_0, r_0 = a_0/5 and r_0 = 5a_0, where a_0 is the Bohr radius is.
(b) Qualitatively describe your expectation of how the shielding will affect the bound-state energies as a function of n, ? and m for the different values ??of r_0. In particular, discuss the question of degeneracy and the number of bound states.
Expert Answer
Answer:To determine the parameters ? and V? in order to smoothly merge the two potentials, we need to match their values and derivatives at . Taking the derivative of with respect to r: At : Taking the derivative of V(r) with respect to r: Matching the values and derivatives at r = r?, we have: Solving these equations simultaneously, we can find the values of ? and V?