4. A particle of mass m moves in a three-dimensional spherically symmetric well, where V = -Vo, r≤a and V = 0, r > a. (a) Starting from the radial equation (derived from the Schrödinger equation): d dr (d) — - 2mr2 h? - [V(r) − E] R = 1(1+1)R. dR dr Find the appropriate equations for u(r) = rR(r), both inside and outside of the well. (b) For states with = 0, show that the solution inside the well is of the form u(r) = Bsin(kr), justify any boundary conditions that you use to arrive at this solution. (c) For the same 1 = 0 states, find the exponential solution, in terms of a constant K, outside the well. Justify any boundary conditions that you use to arrive at this solution. For the bound states of interest here E < 0. (d) Show that the energies of those states with quantum number 1 = 0) are determined by the condition k cot ka = -K. (e) Show that there are no bound states (E < Vo) unless; Explain your reasoning. h²² Vo 8ma²

4. A particle of mass m moves in a three-dimensional spherically symmetric well, where V = -Vo, r≤a and V = 0, r > a. (a) Starting from the radial equation (derived from the Schrödinger equation): d dr (d) — - 2mr2 h? - [V(r) − E] R = 1(1+1)R. dR dr Find the appropriate equations for u(r) = rR(r), both inside and outside of the well. (b) For states with = 0, show that the solution inside the well is of the form u(r) = Bsin(kr), justify any boundary conditions that you use to arrive at this solution. (c) For the same 1 = 0 states, find the exponential solution, in terms of a constant K, outside the well. Justify any boundary conditions that you use to arrive at this solution. For the bound states of interest here E < 0. (d) Show that the energies of those states with quantum number 1 = 0) are determined by the condition k cot ka = -K. (e) Show that there are no bound states (E < Vo) unless; Explain your reasoning. h²² Vo 8ma²