Question 14.4: Show that the states of a p electron, Y11, Y10, and Y1−1, tr...
Show that the states of a p electron, Y_{11}, Y_{10}, and Y_{1−1}, transform according to a three-dimensional representation of the rotation group R(3).
According to Eq. (14.52), the product of the rotation operator pδ(\phi _{z}) and a state Y_{l{m_l}} is
p_{z}(δ\phi _{z}) = 1 − iδ\phi _{z}l_{z}. (14.52)
p_{z}(δ\phi _{z})Y_{l{m_l}} = (1 − i δ\phi _{z}l_{z})Y_{l{m_l}} = (1 − i δ\phi _{z}m_{l})Y_{l{m_l}}
The product of p_{z} (δ\phi _{z}) or l_{z} times a spherical harmonic gives a constant times a spherical harmonic with the same value l and the same value of m_{l}.
According to Eq. (14.53), the product of p_{x} (δ\phi _{x} ) times a spherical harmonic Y_{l{m_l}} is (1 − i δ \phi _{x} l_{x})Y_{l{m_l}} . Then according to Eq. (14.60) and Eqs. (14.58) and (14.59), we get spherical harmonics with the same value of l and one value greater or one value less of the azimuthal quantum number m. The same can be said of the product of p_{y} (δ\phi_{y} ) times a spherical harmonic.
p_{x} (δ\phi_{x} ) = 1 − iδ\phi_{x} l_{x} , (14.53)
l_{-}Y_{l{m_l}} = \sqrt{l(l + 1) − m_{l}(m_{l} + 1)}Y_{l_{ml}-1} (14.58)
l_{+}Y_{l{m_l}} = \sqrt{l(l + 1) − m_{l}(m_{l} + 1)}Y_{l_{ml}+1} (14.59)
l_{x} = \frac{1} {2} (l_{+} + l_{−}). (14.60)
In each case, an infinitesimal rotation of the spherical harmonics of a p-electron gives linear combinations of p-electron states