Homework 1 10310IPT 599000 / Ultrafast Soft X-Rays and Extreme Ultraviolet Radiation Principles and applications / Review Hydrogen atom 1. The normalized ground-‐state wavefunction for the electron in the hydrogen atom is 𝜓 𝑟, 𝜃, 𝜙 = ! ! ! ( )! 𝑒 !!/!! , where r is the radial coordinate of the ! ! ! electron and a0 is the Bohr radius. (a) Sketch the wavefunction versus r. (b) Show that the probability of finding the electron between r and r+dr is given by |𝜓(𝑟)|! 4𝜋𝑟 ! 𝑑𝑟. (c) Sketch the probability versus r and from your sketch find the radius at which the electron is most likely to be found. (d) Show that the wavefunction as given is normalized (e) Find the probability of locating the electron between 𝑟! = 𝑎! /2 and 𝑟! = 3𝑎! /2. 2. Prove that the nth energy level of an atom has degeneracy equal to n2. 3. Compute the probability that a 2s electron of hydrogen will be found inside the Bohr radius for this state. Compare this with the probability of finding a 2p electron in the same region. 4. The general proof of the selection rule is beyond our scope, but you can prove it in a few special cases. (a) It is a fact that the wave function ψ nlm (r ) = Rnl (r )Θlm (θ )eimφ satisfies ψ nlm (−r ) = (−1)lψ nlm (r ) . (1) [This property is often described by saying that the wave function has parity (−1)l .] Use the angular functions listed below to prove (1) for all wave functions with l = 0, 1, or 2. (b) The probability of a radiative transition (n, l , m → nʹ′, l ʹ′, mʹ′) is given by (2), which now takes the form 2 P(n, l , m → nʹ′, l ʹ′, mʹ′) ∝ E0 2 ∫ψ * nʹ′l ʹ′mʹ′ xψ nlm dV . (2) (This is for radiation polarized with E0 in the x direction. For isotropic unpolarized radiation, we must average over this and the two corresponding expressions with x replaced by y and by z.) Use the property (1) to prove that the transition probabilty (2) is zero if l ʹ′ = l (whether the integrand is x, or y, or z). This proves that transitions with Δl = 0 are forbidden. The rule is easy to understand if we recall that the photon has spin 1. This means that when a photon is emitted by or absorbed into an atom, it must change the atom’s angular momentum by 1 unit. 5. The outermost (valence) electron of sodium is in a 3s state when the atom is in its ground state. The valence electron can be excited to higher levels, the first few of which are shown below. Given the selection rule that only those transitions for which Δl = ±1 are allowed, indicate on this energy-level diagram all allowed transitions among the levels shown. l =0 l =1 l =2