Question

2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write any eigenstate as Ψη1m(r) = Rnl (r)Ym(θ, φ) (a) Using the relation L2 Yim( , ф)-n°1(1+1)Yim(9, ф), obtain the differential equation satisfied by Rl). Why does Rnl not depend on the quantum number m? (b) The ground-state wave function for the hydrogen atom has Rio(r)--e-r/ao, Yoo(94)=. (2.3) Va Show by direct insertion into the Schrödinger equation (2.1) that this is indeed an eigenstate of the Hamiltonian, and determine its eigenvalue. (c) Using Eq. (2.3), calculate the average distance between the electron and the proton in the ground state. What would we obtain if we instead calculated the average position r of the electron relative to the proton? (d) Suppose the hydrogen atom has the following real wave function at time t0 (2.4) what is Ψ(r,t) for all time t? Determine the average position r of the electron relative to the proton in this state. Hence, obtain the average electric dipole moment for this state of the hydrogen atom

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