Angular Momentum of Electron in Hydrogen

The spherical harmonics are defined as $Y^m_{\ell}$ where $m$ is the $z$ -axis angular momentum quantum number and $\ell$ is the total angular momentum quantum number. These quantum numbers describe the eigenvalues of the spherical harmonics with respect to the corresponding operators given in the question.

The spherical harmonics given share $m = -1$ . Therefore their sum is still an eigenfunction of the $z$ -axis angular momentum operator, with eigenvalue $-\hbar$ , since both spherical harmonics are multiplied by the same eigenvalue when the operator is applied. However, each takes different values of $\ell$ . When the total angular momentum operator is applied, each gains a different numerical coefficient which cannot be factored out:

$\hat{L}^2 (Y^{-1}_1 + Y^{-1}_2) = 2\hbar^2 (Y^{-1}_1 + 3Y^{-1}_2).$

The expression on the right is not a multiple of $Y^{-1}_1 + Y^{-1}_2$ . Therefore this state is not an eigenstate of the total angular momentum operator.

operator is not linear combination and momentum can superimpose as a resultant vector of them net probability along mutual perpendicular axis.