4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force between the electron and the proton is modeled as a spring. Then V(r)- (1/2)mu22, where m is the mass of the electron and w is the natural frequency of oscillation You're goal is to determine the eigenenergies of the electron and the corresponding wave functions, as outlined below Let's again start with the radial equation associated with the Schrodinge equation 4.37 in Griffiths [where u(r) R(r) ir equation as given by (a) (1 point) Let's define the dimensionless variable -ry/mw/h. Re-express the radial (b) (2 points) Determine the limiting cases of this differential equation [what is u(p) for ρ » 1 (c) (5 points) Define a new function L(p) through the relationship u(p)-L(p)u-(p)u+(p) equation in terms of u(p) (call this u+(p)) and what is u(p) for ρ 1 (call this u-(p))?] Determine the new differential equation for L(p) and use the power series method to solve d) (3 points) Determine the eigenenergies of the electron (How can you truncate your power e) (3 points) What is the total (non-normalized) wave function of the electron in the ground (f) (1 points) What is the degeneracy (number of different spatial quantum states associated it (assume L(p)-Σχ0Ak ρk). What is the recursion relationship? serieS state? with the same energy) for the first excited state? 4. Consider the time-independent Schrödinger equation for an "atom" in which the attractive force between the electron and the proton is modeled as a spring. Then V(r)- (1/2)mu22, where m is the mass of the electron and w is the natural frequency of oscillation You're goal is to determine the eigenenergies of the electron and the corresponding wave functions, as outlined below Let's again start with the radial equation associated with the Schrodinge equation 4.37 in Griffiths [where u(r) R(r) ir equation as given by (a) (1 point) Let's define the dimensionless variable -ry/mw/h. Re-express the radial (b) (2 points) Determine the limiting cases of this differential equation [what is u(p) for ρ » 1 (c) (5 points) Define a new function L(p) through the relationship u(p)-L(p)u-(p)u+(p) equation in terms of u(p) (call this u+(p)) and what is u(p) for ρ 1 (call this u-(p))?] Determine the new differential equation for L(p) and use the power series method to solve d) (3 points) Determine the eigenenergies of the electron (How can you truncate your power e) (3 points) What is the total (non-normalized) wave function of the electron in the ground (f) (1 points) What is the degeneracy (number of different spatial quantum states associated it (assume L(p)-Σχ0Ak ρk). What is the recursion relationship? serieS state? with the same energy) for the first excited state?