Show that the states of a p electron, Y11, Y10, and Y1−1, transform according to a three-dimensional representation of the rotation group R(3).

Question

Show that the states of a p electron, [latex]Y_{11}, Y_{10}[/latex], and [latex] Y_{1−1}[/latex], transform according to a three-dimensional representation of the rotation group R(3).

Accepted Answer

According to Eq. (14.52), the product of the rotation operator [latex] pδ(\phi _{z})[/latex] and a state [latex]Y_{l{m_l}}[/latex] is

[latex]p_{z}(δ\phi _{z}) = 1 − iδ\phi _{z}l_{z}.[/latex] (14.52)

[latex]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}}[/latex]

The product of [latex]p_{z} (δ\phi _{z})[/latex] or [latex]l_{z}[/latex] times a spherical harmonic gives a constant times a spherical harmonic with the same value l and the same value of [latex]m_{l}[/latex].

According to Eq. (14.53), the product of [latex]p_{x} (δ\phi _{x} )[/latex] times a spherical harmonic [latex]Y_{l{m_l}}[/latex] is [latex](1 − i δ \phi _{x} l_{x})Y_{l{m_l}}[/latex] . 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 [latex] p_{y} (δ\phi_{y} )[/latex] times a spherical harmonic.

[latex]p_{x} (δ\phi_{x} ) = 1 − iδ\phi_{x} l_{x} ,[/latex] (14.53)

[latex]l_{-}Y_{l{m_l}} = \sqrt{l(l + 1) − m_{l}(m_{l} + 1)}Y_{l_{ml}-1}[/latex] (14.58)

[latex]l_{+}Y_{l{m_l}} = \sqrt{l(l + 1) − m_{l}(m_{l} + 1)}Y_{l_{ml}+1}[/latex] (14.59)

[latex]l_{x} = \frac{1} {2} (l_{+} + l_{−}).[/latex] (14.60)

In each case, an infinitesimal rotation of the spherical harmonics of a p-electron gives linear combinations of p-electron states

Question 14.4: Show that the states of a p electron, Y11, Y10, and Y1−1, tr...

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