9. Consider 3-dimensional harmonic oscillator (a) Write the Hamiltonean in cartesian and polar coordinates (b) Write the time-independent Schrodinger equation in spherical coordinates. (c) Using the method of separation of variables write down equations satisfied by an- gular functions (#) and ().Find the angular eigenfunctions.(Hint: This part of the problem does not involve any work, you should see that the solution is the same as the one we found for the hydrogen atom.) (d) Write down the equation satisfied by the radial wavefunction. (e) The Hamiltonean depends on 3 dimensionful quantities, A, m, and the frequency w. Use them to construct a quantity p with units of length. Change the variables from the dimensionful radius r to a dimensionless variable y = r/p and write down the radial equation in the new variables. You should also replace the energy E by a dimensionless parameter = E/(hw). (f) Consider a limit of large y and find an approximate solution of f;(y) the radial equation valid in that region (by only keeping terms that are largest at large y). (g) Consider a limit of small y and find an approximate solution f,(y) of the radial equation valid in that region (by only keeping terms that are largest at small y). (h) Write the solution in the form R(y) = fs(y)/fi(y)w(y) and find the equation that w(y) must satisfy for R(y) to be the solution of the radial equation. (i) Look for the solution of the equation for w(y) in the form of a polynomial o0 w(y) = Z g n=0 The solution must be such that the polynomial terminates (otherwise the solution won't be normalizable). (j) Write down the equation (or a set of equations) that the the coefficients , must satisfy. Show that , (and as a consequence , for all odd values of n) must be 0. (k) Reorder the equation to obtain a recursive relation for coefficients c,. Find the allowed values of using the requirement that the polynomial must terminate