44 The hydrogen atom
(a) Show that Θ(θ) satisfies the equation
1
Θ(θ)∂Θ(θ)
∂θ
=mmax
cosθ
sinθ
.(b) Find the solution of the equation for Θ(θ).
(Both sides have the formf′(θ)/f(θ)whose
integral is ln{f(θ)}.) By substituting
this solution into eqn 2.5 to show that
b= mmax(mmax+ 1), or otherwise, obtain
eqn 2.10.(2.12)Parity and selection rules
Show that eqn 2.42 implies thatl 2 −l 1 is odd.
Hence, or otherwise, prove thatIangis zero unless
the initial and final states have opposite parity.(2.13)Selection rules in hydrogen
Hydrogen atoms are excited (by a pulse of laser
light that drives a multi-photon process) to a spe-
cific configuration and the subsequent spontaneous
emission is resolved using a spectrograph. Infra-
red and visible spectral lines are detectedonlyat
the wavelengths 4. 05 μm, 1. 87 μmand0. 656 μm.
Explain these observations and give the values of
nandlfor the configurations involved in these
transitions.Web site:
http://www.physics.ox.ac.uk/users/foot
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