•• The probability of a radiative transition (n, l, m → n′, l′, m′) induced by unpolarized isotropic radiation is given by an average of (11.52) and the two corresponding expressions with x replaced by y and by z. Prove the selection rule that this probability is zero unless m′ = m or m ± 1 (that is, Δm = 0 or ±1). [Hint: Use the form (8.98) of the wave functions and write the integral of (11.52) in terms of spherical polar coordinates. You will need to examine only the integral over ϕ.]
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