*Problem 4.38 Consider the three-dimensional harmonic oscillator, for which the potential is. V (r) = =mw2,2 [4.188] (a) Show that separation of variables in cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer: En = (n + 3/2) hw. [4.189] (b) Determine the degeneracy d(n) of En.2. Griffiths 4.39 * * *Problem 4.39 Because the three-dimensional harmonic oscillator potential (Equa- tion 4.188) is spherically symmetric, the Schrodinger equation can be handled by separation of variables in spherical coordinates, as well as cartesian coordinates. Use the power series method to solve the radial equation. Find the recursion formula for the coefficients, and determine the allowed energies. Check your answer against Equation 4.189. (b) The Cartesian and spherical decompositions give us two different energy eigenbases. Check that the degeneracies of each energy level are consistent between them. (c) For the lowest three energy levels, write the Cartesian eigenstates (x, ny, nz) as linear combinations of the spherical eigenstates (n, (, m) and vice-versa (i.e. find the change-of-basis matrices for the first three energy levels)