10. (50 pts) A particle confined to the surface of a sphere is in the state Where π 2 4(0, 0) = {NG²==0)². N = π5 8 πT 0 < 0 < 2 Π 0, << π 2 -1/2 - 6π³ +48л a. Find the coefficients (including a numerical approximation) for the |l, m) = |1,0), 1,1) terms in a spherical harmonic expansion of (0, 4). = |0,0), |1,-1), b. A student claims that this state has zero angular momentum in the z direction. Is this correct? Justify and explain. c. What is the probability that the measurement of the energy of the particle will yield h²/1? d. Generate a 3D plot of (0, 4), with the value of acting as the radius r in the plot, as a function of spherical coordinates 0, (using something like SphericalPlot3D in mathematica). e. Consider the expansion of (0, 4) in terms of spherical harmonics. Using the first 14 non- zero terms in the expansion, generate seven 3D plots of the expansion like in part d. The seven plots: the first plot includes only the first two terms in the expansion, the second plot includes the first four terms, the third plot includes the first six terms, etc. f. Plot the values of the C10 coefficients found in part e. Are there any local maxima? g. What is the probability that the particle can be found in the region 0 < 0 <π/6? What about л/3<<π/2? On one plot, plot (0,0) and your approximation from part (a) as a function of and check to see if your answer for the probabilities makes sense.