4. The radial part of the Schrödinger equation for a system of two particles, of charges +e and -e, interacting through a Coulombic potential is: -h² d 2μr² dr 2 dR(r) e² R(r) h2 + dr Απερτ 2μr2 l(l+1)R(r) = ER(r). (a) Show that the function: 1 R(r): = 4√2π (+1) 32² (2- - ao r exp 200 with 0, is a solution of the radial Schrödinger equation with the energy eigenvalue E = - нег 8h2 (4πeo)2 Take ao h² (4πEO) μελ (b) Find the expectation value of the radial separation between the two parti- cles. (c) Briefly explain what the physical significance of this expectation value is. (d) A muonic atom consists of a proton and a muon, which has the same charge as the electron. What is the value of μ the reduced mass for this system. You may take the mass of the proton mp = 1.67 × 10-27 kg and the mass of the muon to be mμ = 1.88 × 10-28 kg. (e) Given that the energy levels of the hydrogen atom scale as 12 and that the principal quantum number for the state considered in part(a) is n = 2, what is wavelength of a transition between the n = 2 state and a state with n = 3?

4. The radial part of the Schrödinger equation for a system of two particles, of charges +e and -e, interacting through a Coulombic potential is: -h² d 2μr² dr 2 dR(r) e² R(r) h2 + dr Απερτ 2μr2 l(l+1)R(r) = ER(r). (a) Show that the function: 1 R(r): = 4√2π (+1) 32² (2- - ao r exp 200 with 0, is a solution of the radial Schrödinger equation with the energy eigenvalue E = - нег 8h2 (4πeo)2 Take ao h² (4πEO) μελ (b) Find the expectation value of the radial separation between the two parti- cles. (c) Briefly explain what the physical significance of this expectation value is. (d) A muonic atom consists of a proton and a muon, which has the same charge as the electron. What is the value of μ the reduced mass for this system. You may take the mass of the proton mp = 1.67 × 10-27 kg and the mass of the muon to be mμ = 1.88 × 10-28 kg. (e) Given that the energy levels of the hydrogen atom scale as 12 and that the principal quantum number for the state considered in part(a) is n = 2, what is wavelength of a transition between the n = 2 state and a state with n = 3?