Assignment 8 Q1 0.0/9.0 points (graded) Consider an electron inside a spherical box of radius ry, where we take the potential to be zero inside the box, and we presume the potential is infinitely high outside the sphere. Therefore, we presume that the wavefunctions of any energy eigensolutions have to be zero on the boundary of the sphere. We assume also that the wavefunction solution of this problem can be separated into a product (r) = %x (r)Y (8, ), wherer, @, and , have their usual meanings in spherical polar coordinates. We also presume that y (I:I) = 0 since otherwise the wavefunction would become infinite at r = 0, and we presume that there are some solutions to the radial behavior that are indexed by some integer quantum number 7. In such solutions, we also presume that y (rg] = 0 because of the infinitely high potential barrier at 7. The electron mass is myg and other symbols have their usual meanings. Nate: it is possible to solve this problem for energy eigenstates analytically quite completely in terms of specific functions, and you should start out as if you are going to solve this problem of a particle in a spherical "box', using a similar approach to that for the relative motion Schroedinger equation for the hydrogen atom. Once you have set up the various equations you would have to solve, then consider the following questions. Note that you should not need to perform a full solution of those differential equations to answer the questions below, and you should therefore not need to use any special functions beyond those already discussed in the course. Hint: For many parts of this problem, though this is obviously not the same problem as the hydrogen atom and we should expect some differences, as a starting point, look particularly at the equation forx (r] that arises in the solution of the hydrogen atom problem. Consider whether each of the following six statements is true or false. Note: This is a compound true/false question. In the six statements below, exactly three are true and three are false. For each attempt you make to answer this question, select only the three correct statements. Be sure the other three boxes are not checked. You may want to keep a note of your answer attempts. Because this is a more substantial and challenging question, we have given you a relatively large number of attempts to solve it. |: In the energy eigensolutions of this problem, the angular parts of the solution will be spherical harmonic functions. The lowest energy solution of this problem will correspond to a y (r) that is the same for all r. |: The energy eigen solutions of this problem can be indexed with integer guantum numbers (e.g. i, [, and m), and to know the eigenenergies we need to know the values of all three numbers. ) 2 For [ = 0, there is a solution with eigenenergy Ej, = % (1) . '0 o ) 2 For I = 0, the lowest eigenenergy has a value E, = = (L) 2my |: For [ = 0, the wavefunction corresponding to the lowest eigenenergy can be written in the (unnormalized) form sin(mr/ry) Y(r) =