= Solution
Linearity and the shared $l=1$ angular <eigenvalue> give
$$
\boxed{\hat l^2\psi_x=2\psi_x,\qquad\hat l^2\psi_y=2\psi_y.}
$$
Using the relative phases in part (c),
$$
\boxed{\hat l_z\psi_x=i\psi_y,\qquad\hat l_z\psi_y=-i\psi_x.}
$$
Thus each real orbital remains an <energy eigenstate> with \b[$E=-1/8$ hartree] and has definite <orbital angular momentum> magnitude \b[$\sqrt2\hbar$], but neither is an <eigenfunction> of $\hat l_z$. A measurement of $l_z$ gives \b[$+\hbar$ or $-\hbar$, each with <probability> $1/2$]. Its mean is zero and its variance is $\hbar^2$; a zero mean does not mean that a measurement gives zero.
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