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Problem 1 Radial part of the wave function of hydrogen electrons 9 (3+2+1+3) points The wave function of an electron in the hydrogen atom can be separated into an angular part F(0,4) and a radial part R(r) (see homework V, problem 4). We will have a closer look at the latter. (a) The solution R(r) of the differential equation of the radial component of an electron in the Coulomb potential in the limit r→∞ and for energy E <0 is R(r) = Ber with B being a constant defined by normalisation and with = √√-2moE/h. Thus make the following ansatz for the general solution: R(r) = u(r) e-r. Write down the resulting differential equation for u(r). Use a power series in r to solve this equation and show that the following recursion relation holds for the coefficients: kj-a bj = 2bj-11 ¹j (j+1) - l (l+1) with a := moZe²/47e0h². (b) The power series can only have a finite number of terms in order to be normalizable. Let b₁ = 0 be the first vanishing coefficent. What does that imply for the energy? (c) The radial part of the wave function is known as Laguerre polynomials. Sketch those for n = 1,2. (d) Show that the 1s, 2p and 3d orbitals of a hydrogen-like atom show a single maximum in the radial probability curves. Obtain the values at which these maxima occur.